Answer to Question #246580 in Statistics and Probability for smilynne

Question

Use the traditional method in testing the hypothesis in the problems below. In each problem, state the following:

a. State the hypotheses and identify the clam.

b. Find the critical value(s)

c. Find the test value

d. Make the decision

e. Summarize the result

First graders in the state of Virginia get an average score of 20 on a reading test (higher score reflect higher levels of performance). A teacher is using a new method to teach reading. She predicts that by the end of the first grade, students getting her new method will have significantly higher scores on reading than those in the population. The mean of the 25 students in her class is 23.2 and the standard deviation of the students in the class is 4.7. Do the data provide support for the teacher’s prediction? 

Answer 1

a. The following null and alternative hypotheses need to be tested:

$H_0: \mu\leq20$

$H_1:\mu>20$

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

b. Based on the information provided, the significance level is $\alpha=0.05, df=25-1=24$ degrees of freedom and the critical value for a right-tailed test is $t_c=1.710882.$

The rejection region for this two-tailed test is $R = \{t: t> 1.710882\}.$

c. The z-statistic is computed as follows:

$$t=\dfrac{x-\mu}{s/\sqrt{n}}=\dfrac{23.2-20}{4.7/\sqrt{25}}=3.4043$$

d. Since it is observed that $t=3.4043>1.710882=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value for right-tailed $\alpha=0.05, df=n-1=24$ degrees of freedom is $p=0.001166,$ and since $p=0.001166<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

e. Therefore, there is not enough evidence to claim that the population mean $\mu$ is greater than $20,$ at the $\alpha = 0.05$ significance level.