Q.93 - The Practice of Statistics for AP

Question

Is it significant? For students without special preparation, SAT Math scores in recent years have varied Normally with mean $\mu =518$. One hundred students go through a rigorous training program designed to raise their SAT Math scores by improving their mathematics skills. Use your calculator to carry out a test of

$H_0:\mu =518$

$H_{\alpha }:\mu >518$

in each of the following situations.

(a) The students' scores have mean $\overline{x}=536.7$ and standard deviation $s_x=114$. Is this result significant at the$5\%$ level?

(b) 'The students' scores have mean $\overline{x}=537.0$ and standard deviation $s_x=114$. Is this result significant at the $5\%$ level?

(c) When looked at together, what is the intended lesson of (a) and (b)?

Step-by-Step Solution

Answer

(a) No, it is not significant

(b)Yes, it is significant

(c) The small change in the value of the sample mean could change the result of overall hypothesis testing.

Part (a) Step 1: Given Information

Given in the question that,

Sample mean$\left(\overline{x}\right)=536.7$

Sample standard deviation$\left(s_x\right)=114$

Sample size$\left(n\right)=100$

The hypothesis are:

$H_0:\mu =518$

$H_{\alpha }:\mu >518$

We have to check Is this result significant at the$5\%$ level.

Part (a) Step 2: Explanation

The obtained Ti-83 output is:

The $p$-value is more than the significance level. Thus, the null hypothesis cannot be rejected.

At $5\%$significance level, there are insufficient evidence to support the provided claim that the test is significant.

Part (b) Step 1: Given Information

Given in the question that,

Sample mean$\left(\overline{x}\right)=537.0$

Sample standard deviation$\left(s_x\right)=114$

Sample size$n=100$

The hypothesis are:

$H_0:\mu =518$

$H_{\alpha }:\mu >518$ we have to check Is this result significant at the$5\%$ level.

Part (b) Step 2: Explanation

The obtained Ti-83 output is:

The p-value is less than the significance level. Thus, the null hypothesis can be rejected.

At $5\%$ significance level, there are sufficient evidence to support the provided claim that the test is significant.

Part (C) Step 1: Given Information

We have to find out that  When looked at together, what is the intended lesson of (a) and (b) 

Part (c) Step 2: Explanation

From above parts, it is known that all values are same except the values of the mean, Thus, it could be said that the small change in the value of the sample mean could change the result of overall hypothesis testing.