Answer to Question #346268 in Statistics and Probability for John Lloyd

Question

2. Don, a canteen owner claims that the average meal cost of his usual costumers

is 190 pesos. In order to test his claim, Don took a random sample of 25

costumers and found out that the meal cost is 210 with a standard deviation of

30 pesos. Test the hypothesis at 0.01 level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=190$

$H_1:\mu\not=190$

This corresponds to a two-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 $\alpha = 0.01,$ $df=n-1=24$ and the critical value for a two-tailed test is $t_c =2.79694.$

The rejection region for this two-tailed test is $R = \{t:|t|>2.79694\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{210-190}{30/\sqrt{25}}=3.333333$$

Since it is observed that $|t|=3.333333>2.79694=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, $df=24$ degrees of freedom, $t=3.333333$ is $p=0.002776,$ and since $p=0.002776<0.01=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$

is different than 190, at the $\alpha = 0.01$ significance level.