Answer to Question #139569 in Statistics and Probability for Lin

Question

a) A university wishes to estimate the average number of hours students spend doing homework per week. The variance obtained from previous study is 25 hours. How large a sample must be selected if they want to be 98% confident of finding the true mean given the error is within 1.5 hours?
(b) Find t value for a 95% confidence interval when the sample size is 8.

Answer 1

a) If $\sigma$ not known:

If $n$ is large, then use normal.

If $n$ is small, then use t-distribution.

Margin of error

$$ME=t_{\alpha/2, df}\dfrac{s}{\sqrt{n}}$$

Given $ME=1.5\ h, s=25\ h, \alpha=0.02$

If $n\leq30$

$$ME=t_{\alpha/2, df}\dfrac{25}{\sqrt{n}}>1/5$$

Hence we can use normal distribution

$$ME=z_{\alpha/2}\dfrac{s}{\sqrt{n}}$$

$$z_{\alpha/2}\dfrac{25}{\sqrt{n}}\leq1.5$$

$$n\geq(\dfrac{25z_{\alpha/2}}{1.5})^2$$

$$z_{0.02/2}=2.3263$$

$$n\geq(\dfrac{25\cdot2.3263}{1.5})^2$$

$$n\geq1504$$

(b)

$$t_{\alpha/2, df}=t_{0.05/2, 8-1}=2.364619$$