52 results

distribution - value range for a variable

A compact disc is designed to last an average of 4 years with a standard deviation of 0.8 years. Wha

A compact disc is designed to last an average of 4 years with a standard deviation of 0.8 years. What is the probability that a CD will last less than 3 years?
Using our [URL='http://www.mathcelebrity.com/probnormdist.php?xone=3&mean=4&stdev=0.8&n=1&pl=P%28X+%3C+Z%29']Z-score and Normal distribution calculator[/URL], we get:
[B]0.10565[/B]

A financial analyst computed the ROI for all companies listed on the NYSE. She found that the mean o

A financial analyst computed the ROI for all companies listed on the NYSE. She found that the mean of this distribution was 10% with standard deviation of 5%. She is interested in examining further those companies whose ROI is between 14% and 16% of the approximately 1,500 companies listed on the exchange, how many are of interest of her?
First, use our [URL='http://www.mathcelebrity.com/zscore.php?z=p%280.14%3Cz%3C0.16%29&pl=Calculate+Probability']z-score calculator[/URL] to get P(0.14 < z < 0.16) = 0.007889
Divide that by 2 for two-tail test to get0.003944729
Use the NORMSINV(0.003944729) in Excel to get the Z value of 2.66
Therefore, the companies of interest are 2.66 * 1500 * 0.10 = [B]399[/B]

A group of students at a school takes a history test. The distribution is normal with a mean of 25,

A group of students at a school takes a history test. The distribution is normal with a mean of 25, and a standard deviation of 4.
(a) Everyone who scores in the top 30% of the distribution gets a certificate.
(b) The top 5% of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state?
(a) [URL='http://www.mathcelebrity.com/zcritical.php?a=0.70&pl=Calculate+Critical+Z+Value']Top 30% is 70% percentile[/URL]
Inverse of normal distribution(0.7) = -0.5244005
Plug into z-score formula, -0.5244005 = (x - 25)/4
[B]x = 22.9024[/B]
(b) [URL='http://www.mathcelebrity.com/zcritical.php?a=0.95&pl=Calculate+Critical+Z+Value']Top 5% is 95% percentile[/URL]
Inverse of normal distribution(0.95) = 1.644853627
Plug into z-score formula, 1.644853627 = (x - 25)/4
[B]x = 31.57941451[/B]

A random sample of 25 customers was chosen in CCP MiniMart between 3:00 and 4:00 PM on a Friday afte

A random sample of 25 customers was chosen in CCP MiniMart between 3:00 and 4:00 PM on a Friday afternoon. The frequency distribution below shows the distribution for checkout time (in minutes).
Checkout Time (in minutes) | Frequency | Relative Frequency
1.0 - 1.9 | 2 | ?
2.0 - 2.9 | 8 | ?
3.0 - 3.9 | ? | ?
4.0 - 5.9 | 5 | ?
Total | 25 | ?
(a) Complete the frequency table with frequency and relative frequency.
(b) What percentage of the checkout times was less than 3 minutes?
(c)In what class interval must the median lie? Explain your answer.
(d) Assume that the largest observation in this dataset is 5.8. Suppose this observation were incorrectly recorded as 8.5 instead of 5.8. Will the mean increase, decrease, or remain the same? Will the median increase, decrease or remain the same? Why?
(a)
[B]Checkout Time (in minutes) | Frequency | Relative Frequency
1.0 - 1.9 | 2 | 2/25
2.0 - 2.9 | 8 | 8/25
3.0 - 3.9 | 10 (since 25 - 5 + 8 + 2) = 10 | 10/25
4.0 - 5.9 | 5 | 5/25
Total | 25 | ?[/B]
(b) (2 + 8)/25 = 10/25 = [B]40%[/B]
c) [B]3.0 - 3.9[/B] since 2 + 8 + 10 + 5 = 25 and 13 is the middle value which occurs in the 3.0 - 3.9 interval
(d) [B]Mean increases[/B] since it's a higher value than usual. Median would not change as the median is the most frequent distribution and assuming the 5.8 is only recorded once.

A random variable X follows the uniform distribution with a lower limit of 670 and an upper limit

A random variable X follows the uniform distribution with a lower limit of 670 and an upper limit
a. Calculate the mean and standard deviation of this distribution. (Round intermediate calculation for standard deviation to 4 decimal places and final answer to 2 decimal places.)
Using our [URL='http://www.mathcelebrity.com/uniform.php?a=+670&b=+770&x=+680&t=+3&pl=PDF']uniform distribution calculator[/URL], we get:
[B]Mean = 720
Standard deviation = 28.87
[/B]
b. What is the probability that X is less than 730? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
Using our [URL='http://www.mathcelebrity.com/uniform.php?a=+670&b=+770&x=+730&t=+3&pl=CDF']uniform distribution calculator[/URL], we get:
[B]0.6[/B]

An analysis of the final test scores for Managerial Decision Making Tools reveals the scores follow

An analysis of the final test scores for Managerial Decision Making Tools reveals the scores follow the normal probability distribution. The mean of the distribution is 75 and the standard deviation is 8. The instructor wants to award an "A" to students whose score is in the highest 10 percent. What is the dividing point for those students who earn an "A"?
Top 10% is equivalent to the 90th percentile.
Using our [URL='http://www.mathcelebrity.com/percentile_normal.php?mean=+75&stdev=8&p=+90&pl=Calculate+Percentile']percentile calculator[/URL], the 90th percentile cutoff point is [B]85.256[/B]

Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a

Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.
a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?
b. What proportion of the vehicles would be going less than 50 mph?
c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?
d. In what way do you think the actual distribution of speeds differs from a normal distribution?
a. Using our [URL='http://www.mathcelebrity.com/probnormdist.php?xone=65&mean=71&stdev=8&n=+1&pl=P%28X+%3C+Z%29']z-score calculator[/URL], we see that P(x<65) = [B]22.66%[/B]
b. Using our [URL='http://www.mathcelebrity.com/probnormdist.php?xone=+50&mean=71&stdev=8&n=+1&pl=P%28X+%3C+Z%29']z-score calculator[/URL], we see that P(x<50) = [B]0.4269%[/B]
c. [URL='http://www.mathcelebrity.com/zcritical.php?a=0.9&pl=Calculate+Critical+Z+Value']Inverse of normal for 90% percentile[/URL] = 1.281551566
Plug into z-score formula: (x - 71)/8 = 1.281551566
[B]x = 81.25241252[/B]
d. [B]The shape/ trail differ because the normal distribution is symmetric with relatively more values at the center. Where the actual has a flatter trail and could be expected to occur.[/B]

Basic Statistics

Free Basic Statistics Calculator - Given a number set, and an optional probability set, this calculates the following statistical items:

Expected Value

Mean = μ

Variance = σ^{2}

Standard Deviation = σ

Standard Error of the Mean

Skewness

Mid-Range

Average Deviation (Mean Absolute Deviation)

Median

Mode

Range

Pearsons Skewness Coefficients

Entropy

Upper Quartile (hinge) (75th Percentile)

Lower Quartile (hinge) (25th Percentile)

InnerQuartile Range

Inner Fences (Lower Inner Fence and Upper Inner Fence)

Outer Fences (Lower Outer Fence and Upper Outer Fence)

Suspect Outliers

Highly Suspect Outliers

Stem and Leaf Plot

Ranked Data Set

Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range

Root Mean Square

Weighted Average (Weighted Mean)

Frequency Distribution

Successive Ratio

Expected Value

Mean = μ

Variance = σ

Standard Deviation = σ

Standard Error of the Mean

Skewness

Mid-Range

Average Deviation (Mean Absolute Deviation)

Median

Mode

Range

Pearsons Skewness Coefficients

Entropy

Upper Quartile (hinge) (75th Percentile)

Lower Quartile (hinge) (25th Percentile)

InnerQuartile Range

Inner Fences (Lower Inner Fence and Upper Inner Fence)

Outer Fences (Lower Outer Fence and Upper Outer Fence)

Suspect Outliers

Highly Suspect Outliers

Stem and Leaf Plot

Ranked Data Set

Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range

Root Mean Square

Weighted Average (Weighted Mean)

Frequency Distribution

Successive Ratio

Bernoulli Trials

Free Bernoulli Trials Calculator - Given a success probability p and a number of trials (n), this will simulate Bernoulli Trials and offer analysis using the Bernoulli Distribution. Also calculates the skewness, kurtosis, and entropy

Binomial Distribution

Free Binomial Distribution Calculator - Calculates the probability of 3 separate events that follow a binomial distribution. It calculates the probability of exactly k successes, no more than k successes, and greater than k successes as well as the mean, variance, standard deviation, skewness and kurtosis.

Also calculates the normal approximation to the binomial distribution with and without the continuity correction factor

Calculates moment number t using the moment generating function

Also calculates the normal approximation to the binomial distribution with and without the continuity correction factor

Calculates moment number t using the moment generating function

Binominal Probability

You want the binomial distribution, where a "success" is that the plant [U]does not[/U] grow.
So if the probability that the plant grows is 0.9, the probability it does not grow is 1 - 0.9 = 0.1.
We have n = 12, p = 0.1
You want the probability that exactly 4 of 12 do not grow.
Use our [URL='http://www.mathcelebrity.com/binomial.php?n=+12&p=+0.1&k=+4&t=+5&pl=P%28X+%3D+k%29']binomial distribution probability calculato[/URL]r to get P(X = 4) = [B]0.0213[/B]

Chi-Square Critical Values

Free Chi-Square Critical Values Calculator - Given a probability, this calculates the critical value for the right-tailed and left-tailed tests for the Chi-Square Distribution. CHIINV from Excel is used as well.

Compared to the normal distribution, the t distribution has ___ values at the top and ___ at the tai

Compared to the normal distribution, the t distribution has ___ values at the top and ___ at the tails.
a. More; less
b. More; more
c. Less; less
d. Less; more
[B]d. Less; more[/B]
[I]t value chart is wider and flatter[/I]

Confidence Interval for the Mean

Free Confidence Interval for the Mean Calculator - Calculates a (90% - 99%) estimation of confidence interval for the mean given a small sample size using the student-t method with (n - 1) degrees of freedom or a large sample size using the normal distribution Z-score (z value) method including Standard Error of the Mean. confidence interval of the mean

Covariance and Correlation coefficient (r) and Least Squares Method and Exponential Fit

Free Covariance and Correlation coefficient (r) and Least Squares Method and Exponential Fit Calculator - Given two distributions X and Y, this calculates the following:

* Covariance of X and Y denoted Cov(X,Y)

* The correlation coefficient r.

* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)

Exponential Fit

* Coefficient of Determination r squared r^{2}

* Spearmans rank correlation coefficient

* Wilcoxon Signed Rank test

* Covariance of X and Y denoted Cov(X,Y)

* The correlation coefficient r.

* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)

Exponential Fit

* Coefficient of Determination r squared r

* Spearmans rank correlation coefficient

* Wilcoxon Signed Rank test

Critical Z-values

Free Critical Z-values Calculator - Given a probability from a normal distribution, this will generate the z-score critical value. Uses the NORMSINV Excel function.

Customers arrive at the claims counter at the rate of 20 per hour (Poisson distributed). What is th

Customers arrive at the claims counter at the rate of 20 per hour (Poisson distributed). What is the probability that the arrival time between consecutive customers is less than five minutes?
Use the [I]exponential distribution[/I]
20 per 60 minutes is 1 every 3 minutes
1/λ = 3 so λ = 0.333333333
Using the [URL='http://www.mathcelebrity.com/expodist.php?x=+5&l=0.333333333&pl=CDF']exponential distribution calculator[/URL], we get F(5,0.333333333) = [B]0.811124396848[/B]

Eighty percent of the employees at Rowan University have their biweekly Wages deposit directly to th

Eighty percent of the employees at Rowan University have their biweekly Wages deposit directly to their bank by electronic deposit program. Suppose we select a random samples of 8 employees. What is the probability that three of the eight (8) sampled employees use direct deposit program?
Use the [I]binomial distribution[/I]
[LIST]
[*]p = 0.8
[*]n = 8
[*]k = 3
[/LIST]
So we want P(X = 3)
Using our [URL='http://www.mathcelebrity.com/binomial.php?n=+8&p=+0.8&k=+3&t=+5&pl=P%28X+=+k%29']binomial distribution calculator[/URL], we get P(X = 3) = [B]0.0092[/B]

Exponential Distribution

Free Exponential Distribution Calculator - Calculates the Probability Density Function (PDF) and Cumulative Density Function (CDF) of the exponential distribution as well as the mean, variance, standard deviation, and entropy.

Facebook provides a variety of statistics on its Web site that detail the growth and popularity of t

Facebook provides a variety of statistics on its Web site that detail the growth and popularity of the site.
On average, 28 percent of 18 to 34 year olds check their Facebook profiles before getting out of bed in the morning. Suppose this percentage follows a normal distribution with a standard deviation of five percent.
a. Find the probability that the percent of 18 to 34-year-olds who check Facebook before getting out of bed in the morning is at least 30.
b. Find the 95th percentile, and express it in a sentence.
a. P(X >=0.30), calculate the [URL='http://www.mathcelebrity.com/probnormdist.php?xone=+0.30&mean=+0.28&stdev=+0.05&n=+1&pl=P%28X+%3E+Z%29']z-score[/URL] which is:
Z = 0.4
P(x>0.4) = [B]0.344578 or 34.46%[/B]
b. Inverse Normal (0.95) [URL='http://www.mathcelebrity.com/zcritical.php?a=0.95&pl=Calculate+Critical+Z+Value']calculator[/URL] = 1.644853627
Use NORMSINV(0.95) on Excel
0.28 + 0.05(1.644853627) = [B]0.362242681 or 36.22%[/B]

For the normal distribution with parameters μ = 4, σ = 3 ; calculate P(x > 1)

For the normal distribution with parameters μ = 4, σ = 3 ; calculate P(x > 1)
[URL='https://www.mathcelebrity.com/probnormdist.php?xone=1&mean=4&stdev=3&n=1&pl=P%28X+%3E+Z%29']Using our calculator[/URL], we get P(x > 1) = [B]0.841345[/B]

Frequency Distribution Table

Free Frequency Distribution Table Calculator - Determines the classes and frequency distribution using the 2 to k rule.

Geometric Distribution

Free Geometric Distribution Calculator - Using a geometric distribution, it calculates the probability of exactly k successes, no more than k successes, and greater than k successes as well as the mean, variance, standard deviation, skewness, and kurtosis.

Calculates moment number t using the moment generating function

Calculates moment number t using the moment generating function

Hypergeometric Distribution

Free Hypergeometric Distribution Calculator - Calculates the probability of drawing x objects out of a subgroup of k with n possibilities in a total group of N using the hypergeometric distribution.

If approximately 68% of the scores lie between 40.7 and 59.9 , then the approximate value of the sta

If approximately 68% of the scores lie between 40.7 and 59.9 , then the approximate value of the standard deviation for the distribution, according to the empirical rule, is
The empirical rule states 68% of the values lie within 1 standard deviation of the mean. The mean is the midpoint of the interval above:
(59.9 + 40.7)/2 = 50.3
Standard deviation is the absolute value of the mean - endpoint
|59.9 - 50.3| = [B]9.6[/B]

If the distribution of IQ scores is bell-shaped, with a mean of 100 and a standard deviation of 15,

If the distribution of IQ scores is bell-shaped, with a mean of 100 and a standard deviation of 15, then approximately ____% of IQ scores are less than 55?
A bell-shaped curved implies a normal distribution. By using our [URL='https://www.mathcelebrity.com/probnormdist.php?xone=55&mean=100&stdev=15&n=1&pl=Empirical+Rule']empirical rule calculator[/URL], we see that:
99.7% of all normal distribution values lie within 3 standard deviations of the mean.
This means the percent of scores less than 55 which is 3 standard deviations away from the mean is:
100% - 99.7% = [B]0.3%[/B]

Jeff Bezos, who owns Amazon, has a net worth of approximately $143.1 billion (as of mid-2018). An em

Jeff Bezos, who owns Amazon, has a net worth of approximately $143.1 billion (as of mid-2018). An employee in the Amazon distribution center earns about $13 an hour. The estimated lifespan of the employee is 71 years. If the employee worked 24 hours a day, every day of the year from the moment of his birth, how many lifespans would it take for him to earn wages equivalent to Jeff Bezos' net worth? Round the answer to the nearest whole number.
Calculate earnings per lifespan:
Earnings per lifespan = lifespan in years * Annual Earnings
Earnings per lifespan = 71 * 13 * 24 * 365 <-- (24 hours per day * 365 days per year)
Earnings per lifespan = 8,085,480
Calculate the number of lifespans needed to match Jeff Bezos earnings:
Number of lifespans = Jeff Bezos Net Worth / Earnings Per Lifespan
Number of lifespans = 143,100,000,000 / 8,085,480
Number of lifespans = [B]17,698.39 ~ 17,699[/B]

Mimi just started her tennis class three weeks ago. On average, she is able to return 20% of her opp

Mimi just started her tennis class three weeks ago. On average, she is able to return 20% of her opponent's serves. Assume her opponent serves 8 times. Show all work. Let X be the number of returns that Mimi gets. As we know, the distribution of X is a binomial probability distribution.
a) What is the number of trials (n), probability of successes (p) and probability of failures (q), respectively?
b) Find the probability that that she returns at least 1 of the 8 serves from her opponent.
(c) How many serves can she expect to return?
a) [B]n = 8
p = 0.2[/B]
q = 1 - p
q = 1 - 0.2
[B]q = 0.8
[/B]
b) [B]0.4967[/B] on our [URL='http://www.mathcelebrity.com/binomial.php?n=+8&p=0.2&k=1&t=+5&pl=P%28X+>+k%29']binomial calculator[/URL]
c) np = 8(0.2) = 1.6 ~ [B]2[/B] using the link above

Multinomial Distribution

Free Multinomial Distribution Calculator - Given a set of x_{i} counts and a respective set of probabilities θ_{i}, this calculates the probability of those events occurring.

Negative Binomial Distribution

Free Negative Binomial Distribution Calculator - Calculates the probability of the k^{th} success on the x^{th} try for a negative binomial distribution also known as the Pascal distribution.? ? It calculates the probability of exactly k successes, no more than k successes, and greater than k successes as well as the mean, variance, and standard deviation.

Normal Distribution

Free Normal Distribution Calculator - Calculates the probability that a random variable is less than or greater than a value or between 2 values using the Normal Distribution z-score (z value) method (Central Limit Theorem).

Also calculates the Range of values for the 68-95-99.7 rule, or three-sigma rule, or empirical rule. Calculates z score probability

Also calculates the Range of values for the 68-95-99.7 rule, or three-sigma rule, or empirical rule. Calculates z score probability

P-Hat Confidence Interval

Free P-Hat Confidence Interval Calculator - Given a large sized distribution, and a success amount for a certain criteria x, and a confidence percentage, this will calculate the confidence interval for that criteria.

Percentile for Normal Distribution

Free Percentile for Normal Distribution Calculator - Given a mean, standard deviation, and a percentile range, this will calculate the percentile value.

Poisson Distribution

Free Poisson Distribution Calculator - Calculates the probability of 3 separate events that follow a poisson distribution.

It calculates the probability of exactly k successes P(x = k)

No more than k successes P (x <= k)

Greater than k successes P(x >= k)

Each scenario also calculates the mean, variance, standard deviation, skewness, and kurtosis.

Calculates moment number t using the moment generating function

It calculates the probability of exactly k successes P(x = k)

No more than k successes P (x <= k)

Greater than k successes P(x >= k)

Each scenario also calculates the mean, variance, standard deviation, skewness, and kurtosis.

Calculates moment number t using the moment generating function

Random Sampling from the Normal Distribution

Free Random Sampling from the Normal Distribution Calculator - This performs hypothesis testing on a sample mean with critical value on a sample mean or calculates a probability that Z <= z or Z >= z using a random sample from a normal distribution.

Standard Normal Distribution

Free Standard Normal Distribution Calculator - Givena normal distribution z-score critical value, this will generate the probability. Uses the NORMSDIST Excel function.

Student-t Distribution Critical Values

Free Student-t Distribution Critical Values Calculator - Given an α value and degrees of freedom, this calculates the right-tailed test and left-tailed test critical values for the Student-t Distribution

Suppose that previously collected traffic data indicate that, during the afternoon rush hour, an ave

Suppose that previously collected traffic data indicate that, during the afternoon rush hour, an average of 4 cars arrive at a toll bridge each second. If it is assumed that cars arrive randomly, and can thus be modeled with Poisson distribution, what is the probability that in the next second, [U][B]NO[/B][/U] cars will arrive?
Use the [I]Poisson Distribution[/I] with λ = 4 and x = 0
Using the [URL='http://www.mathcelebrity.com/poisson.php?n=+10&p=+0.4&k=+0&t=+3&pl=P%28X+=+k%29']Poisson Distribution calculator[/URL], we get P(0; 4) = [B]0.0183[/B]

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed wit

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.
a. If X = average distance in feet for 49 fly balls, then X ~ _______(_______,_______)

b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X. Shade the region corresponding to the probability. Find the probability.

c. Find the 80^{th} percentile of the distribution of the average of 49 fly balls
a. N(250, 50/sqrt(49)) = [B]0.42074[/B]
b. Calculate Z-score and probability = 0.08 shown [URL='http://www.mathcelebrity.com/probnormdist.php?xone=+240&mean=+250&stdev=+7.14&n=+1&pl=P%28X+%3C+Z%29']here[/URL]
c. Inverse of normal distribution(0.8) = 0.8416. Use NORMSINV(0.8) [URL='http://www.mathcelebrity.com/zcritical.php?a=0.8&pl=Calculate+Critical+Z+Value']calculator[/URL]
Using the Z-score formula, we have
0.8416 = (x - 250)/50
x = [B]292.08[/B]

b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X. Shade the region corresponding to the probability. Find the probability.

c. Find the 80

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed wit

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and
a standard deviation of 50 feet.
a. If X = distance in feet for a fly ball, then X ~
b. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? Sketch the graph. Scale the horizontal axis X. Shade the region corresponding to the probability. Find the probability.
c. Find the 80th percentile of the distribution of fly balls. Sketch the graph, and write the probability statement.
a. [B]N(250, 50/sqrt(1))[/B]
b. Calculate [URL='http://www.mathcelebrity.com/probnormdist.php?xone=+220&mean=250&stdev=50&n=+1&pl=P%28X+%3C+Z%29']z-score[/URL]
Z = -0.6 and P(Z < -0.6) = [B]0.274253[/B]
c. Inverse of normal distribution(0.8) = 0.8416 using NORMSINV(0.8) [URL='http://www.mathcelebrity.com/zcritical.php?a=0.8&pl=Calculate+Critical+Z+Value']calculator[/URL]
Z-score formula: 0.8416 = (x - 250)/50

x = [B]292.08[/B]

x = [B]292.08[/B]

Tabular Display

Free Tabular Display Calculator - Enter a set of x and p(x) in a tabular probability distribution format and this will evaluate if it is valid or not.

The distribution of actual weights of 8 oz chocolate bars produced by a certain machine is normal wi

The distribution of actual weights of 8 oz chocolate bars produced by a certain machine is normal with µ=8.1 ounces and σ=0.1 ounces. A sample of 5 of these chocolate bars is selected. What is the probability that their average weight is less than 8 ounces?
Calculate Z score and probability using [URL='http://www.mathcelebrity.com/probnormdist.php?xone=8&mean=8.1&stdev=0.1&n=5&pl=P%28X+%3C+Z%29']our calculator[/URL]:
Z = -2.236
P(X < -2.236) = [B]0.012545[/B]

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered t

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as Benford's Law. For example, the following distribution represents the first digits in 231 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.
Digit, Probability
1, 0.301
2, 0.176
3, 0.125
4, 0.097
5, 0.079
6, 0.067
7, 0.058
8, 0.051
9, 0.046
[B][U]Fradulent Checks[/U][/B]
Digit, Frequency
1, 36
2, 32
3, 45
4, 20
5, 24
6, 36
7, 15
8, 16
9, 7
Complete parts (a) and (b).
(a) Using the level of significance α = 0.05, test whether the first digits in the allegedly fraudulent checks obey Benford's Law. Do the first digits obey the Benford's Law?

Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

The hourly wages of employees at Rowan have a mean wage rate of $10 per hour with a standard deviati

The hourly wages of employees at Rowan have a mean wage rate of $10 per hour with a standard deviation of $1.20. What is the probability the mean hourly wage of a random sample of 36 employees will be larger than $10.50? Assume the company has a total of 1,000 employees
Using our [URL='http://www.mathcelebrity.com/probnormdist.php?xone=10.5&mean=10&stdev=1.2&n=36&pl=P%28X+>+Z%29']normal distribution calculator[/URL], we get P(x > 10.5) = [B]0.00621[/B]

The IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. a) What i

The IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
a) What is the probability that a randomly person has an IQ between 85 and 115?
b) Find the 90th percentile of the IQ distribution
c) If a random sample of 100 people is selected, what is the standard deviation of the sample mean?
a) [B]68%[/B] from the [URL='http://www.mathcelebrity.com/probnormdist.php?xone=50&mean=100&stdev=15&n=1&pl=Empirical+Rule']empirical rule calculator[/URL]
b) P(z) = 0.90. so z = 1.28152 using Excel NORMSINV(0.9)

(X - 100)/10 = 1.21852 X = [B]113[/B] rounded up c) Sample standard deviation is the population standard deviation divided by the square root of the sample size 15/sqrt(100) = 15/10 =[B] 1.5[/B]

(X - 100)/10 = 1.21852 X = [B]113[/B] rounded up c) Sample standard deviation is the population standard deviation divided by the square root of the sample size 15/sqrt(100) = 15/10 =[B] 1.5[/B]

The patient recovery time from a particular surgical procedure is normally distributed with a mean o

The patient recovery time from a particular surgical
procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.
What is the median recovery time?
a. 2.7
b. 5.3
c. 7.4
d. 2.1
[B]b. 5.3 (mean, median, and mode are all the same in a normal distribution)[/B]

The relief time provided by a standard dose of a popular children’s allergy medicine averages 7.9

The relief time provided by a standard dose of a popular children’s allergy medicine averages 7.9 hours with a standard deviation of 2.2 hours. Use Table 1.
a. Determine the percentage of children who experience relief for less than 6.4 hours if the relief time follows a normal distribution. (Round your answer to 2 decimal places.)
Using our [URL='http://www.mathcelebrity.com/probnormdist.php?xone=6.4&mean=7.9&stdev=2.2&n=1&pl=P%28X+%3C+Z%29']normal distribution calculator[/URL], we get
Answer = [B]0.25[/B]

True or False (a) The normal distribution curve is always symmetric to its mean. (b) If the variance

True or False
(a) The normal distribution curve is always symmetric to its mean.
(b) If the variance from a data set is zero, then all the observations in this data set are identical.
(c) P(A AND A^{c})=1, where A^{c} is the complement of A.
(d) In a hypothesis testing, if the p-value is less than the significance level α, we do not have sufficient evidence to reject the null hypothesis.
(e) The volume of milk in a jug of milk is 128 oz. The value 128 is from a discrete data set.
[B](a) True, it's a bell curve symmetric about the mean
(b) True, variance measures how far a set of numbers is spread out. A variance of zero indicates that all the values are identical
(c) True. P(A) is the probability of an event and P(Ac) is the complement of the event, or any event that is not A. So either A happens or it does not. It covers all possible events in a sample space.
(d) False, we have sufficient evidence to reject H0.
(e) False. Volume can be a decimal or fractional. There are multiple values between 127 and 128. So it's continuous.[/B]

True or False: The standard deviation of the chi-square distribution is twice the mean.

True or False: The standard deviation of the chi-square distribution is twice the mean.
[B]False[/B], the variance is twice the mean. Mean is k, Variance is 2k

Uniform Distribution

Free Uniform Distribution Calculator - This calculates the following items for a uniform distribution

* Probability Density Function (PDF) ƒ(x)

* Cumulative Distribution Function (CDF) F(x)

* Mean, Variance, and Standard Deviation

Calculates moment number t using the moment generating function

* Probability Density Function (PDF) ƒ(x)

* Cumulative Distribution Function (CDF) F(x)

* Mean, Variance, and Standard Deviation

Calculates moment number t using the moment generating function

Which of the following is NOT TRUE about the distribution for averages?

Which of the following is NOT TRUE about the distribution for averages?
a. The mean, median, and mode are equal.
b. The area under the curve is one.
c. The curve never touches the x-axis.
d. The curve is skewed to the right.
Answer is d, the curve is skewed to the right
For a normal distribution:
[LIST]
[*] The area under the curve for a standard normal distribution equals 1
[*] Mean media mode are equal
[*] Never touches the x-axis since in theory, all events have some probability of occuring
[/LIST]

Z Score Lookup

Free Z Score Lookup Calculator - Given a Z-score probability statement from the list below, this will determine the probability using the normal distribution z-table.

* P(z < a)

* P(z <= a)

* P(z > a)

* P(z >= a)

* P(a < z < b) Calculates z score probability

* P(z < a)

* P(z <= a)

* P(z > a)

* P(z >= a)

* P(a < z < b) Calculates z score probability