Question

It is recommended that adults get 8 hours of sleep each night. A researcher hypothesized college students got less than the recommended number of hours of sleep each​ night, on average. The researcher randomly sampled 20 college students and obtained a​ p-value of 0.10. Suppose the researcher sampled more college students and that the sample mean and sample standard deviation stayed the same. Would the​ p-value be​ lower, be​ higher, or stay the​ same?

Answer 1

The p-value becomes lower.

Explanation:

Let
`\mu` (unknown) be the true mean of number of hours of sleep each night. If we have the hypothesis
`H_(0): \mu = \mu_(0)` vs
`H_(1): \mu < \mu_(0)`, the test statistic for a small sample size n = 20 is
`T=\frac{\bar{X}-\mu_(0)}{S/√(n)}`, where
`\bar{X}` and S are the sample mean and standard deviation respectively. If
`H_(0)` is true, we know that T has a t distribution with n-1 degrees of freedom. The p-value for the lower-tail alternative is computed as
`P(T < t_(1))`, where
`t_(1)` is the observed value for the given sample. If we increase the size of the sample and the sample mean and sample standard deviation stayed the same, then, the value
`s/√(n)` becomes smaller, besides, this quantity is always positive. The absolute value of
`\bar{x}-\mu_(0)` stays the same, and the absolute value of the observed value
`t_(2)=\frac{\bar{x}-\mu_(0)}{s/√(n)}` becomes bigger.

If
`t_(1)` is negative, then,
`t_(2)` is also negative and smaller than
`t_(1)`, therefore,
`P(T < t_(2))` becomes lower.

Answer 2

Sampling more College students by the research will lower the p-value.

Explanation:

A larger sample size provides consistent evidence of the non zero effect against the null hypothesis than a smaller sample size. A sample size of 20 students collected by the researcher is prone to random error and bias compare to a larger sample size. A p-value closer to 0 indicates strong evidence against a null hypothesis.