Question

1. the recommended daily calorie intake for teenage girls is 2,200 calories/day. a nutritionist at a state university believes the average daily caloric intake of girls in that state to be lower. test that hypothesis, at the 5% level of significance, against the null hypothesis that the population average is 2,200 calories/day using the following sample data: n=36,x¯=2,150,s=203.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\ge2200$

$H_1:\mu<2200$

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ $df=n-1=35$ and the critical value for a left-tailed test is $t_c =- 2.030108.$

The rejection region for this left-tailed test is $R = \{t:t<- 2.030108\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{2150-2200}{203/\sqrt{36}}=-1.4778$$

Since it is observed that $t=-1.4778>- 2.030108=t_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for left-tailed, $df=35$ degrees of freedom, $t=-1.4778$ is $p=0.074202,$ and since $p=0.074202>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$

is less than 2200, at the $\alpha = 0.05$ significance level.