Answer to Question #220536 in Statistics and Probability for tvx

Question

Given the population heights of 15 000 male students at a college with mean 170 centimeters and standard deviation of 40cm. 100 random samples of 30 students are obtained.

i)    Is the sampling distribution normally distributed? Explain.

ii)    Calculate the mean of sample means and standard error of mean.

iii)    Find the probability that the mean of the samples is between 165 and 175cm.

Answer 1

i) The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough. A sample size of 30 is large enough when the population distribution is roughly bell-shaped.

ii) $\mu_{\bar x}=170.$

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}*\sqrt{\frac{N-n}{n-1}}=\frac{40}{\sqrt{30}}*\sqrt{\frac{1500-30}{1500-1}}=7.23.$

iii) $P(165<X<175)=P(\frac{165-170}{7.23}<Z<\frac{175-170}{7.23})=P(-0.69<Z<0.69)=$

$=P(Z<0.69)-P(Z<-0.69)=0.5098.$