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Answer

↓Steps Explained:↓

Calculate the Z score using the Normal Approximation to the Binomial distribution given:

n = 10 and p = 0.4 with 3 successes

with and without the Continuity Correction Factor

The Normal Approximation to the Binomial Distribution Formula is below:

Calculate nq to see if we can use the Normal Approximation:

Since q = 1 - p, we have n(1 - p) = 10(1 - 0.4)

nq = 10(0.6)

nq = 6

Since np and nq are both not greater than 5, we cannot use the Normal Approximation to the Binomial Distribution.

Calculate the mean μ (expected value)

μ  =  np

μ  =  10 x 0.4

μ = 4

Calculate the variance σ²

σ²  =  np(1 - p)

σ²  =  10 x 0.4 x (1 - 0.4)

σ²  =  4 x 0.6

σ² = 2.4

Calculate the standard deviation σ

σ  =  √σ² = √np(1 - p)

σ  =  √2.4

σ = 1.5492

Plug in values for Z score

Using the mean (np) = 4 and the standard deviation √np(1 - p) = 1.5492 and 3 successes calculated above, we can find our Z score

Z = -0.6455

Calculate P(Z < -0.6455)

Using our z-score calculator for P(Z < -0.6455)

Calculate P(Z > -0.6455)

Using our z-score calculator for P(Z > -0.6455)

Now calculate the probabilities using the Continuity Correction Factor by subtracting 0.5:

Z = -0.9682

Calculate P(Z < -0.9682)

Using our z-score calculator for P(Z < -0.9682)

Calculate P(Z > -0.9682)

Using our z-score calculator for P(Z > -0.9682)

Final Answer