Question

A population of unknown shape has a mean of 75. You select a sample of 20. The standard deviation of the sample is 5. Compute the probability the sample mean is: a. Between 76 and 77. (0.5 Points) b. Greater than 77. (0.5 Points)

Answer 1

Answer: a. 0.1500 b. 0.0367

Step-by-step explanation:

Let X is a random variable with a distribution that may be known or unknown:

`\mu_x` = the mean of X

`\sigma_x` = the standard deviation of X

If we draw random samples of size n, then the random samples contains sample means
`\overline{X}`, tends to be normally distributed

`\overline{X}\sim N(\mu_x,(\sigma_x)/(√(n)))`

Here,
`\mu_x` = 75

`\sigma_x` =5

n=20

a.

`P(76<\overline{X}<77)=P((76-75)/((5)/(√(20)))<\frac{\overline{X}-\mu}{(\sigma)/(√(n))}<(77-75)/((5)/(√(20))))\\\\=P(0.89<Z<1.79) \ \ [z=\frac{\overline{X}-\mu}{(\sigma)/(√(n))}]\\\\=P(z<1.79)-P(z<0.89)\\\\=0.963273-0.813267=0.150006\approx0.1500`

b.

`P(\overline{X}>77)=P(\frac{\overline{X}-\mu}{(\sigma)/(√(n))}>(77-75)/((5)/(√(20))))\\\\=P(Z>1.79) \ \ [z=\frac{\overline{X}-\mu}{(\sigma)/(√(n))}]\\\\=1-P(z<1.79)\\\\=1-0.963273=0.036727\approx0.0367`