Question

A random survey of 1000 students nationwide showed a mean ACT score of 21.1. Ohio was not used. A survey of 500 randomly selected Ohio scores showed a mean of 20.8. The population standard deviation is 3. The goal of the study is to decide if we can conclude that Ohio is below the national average. Use α = 0.1. What would be the hypotheses. Identify the claim. What would be the critical value(s).

Answer 1

Answer:

We conclude that the mean Ohio score is below the national average.

Step-by-step explanation:

We are given that a random survey of 1000 students nationwide showed a mean ACT score of 21.1. Ohio was not used.

A survey of 500 randomly selected Ohio scores showed a mean of 20.8. The population standard deviation is 3.

Let $\mu$ = mean Ohio scores.

So, Null Hypothesis, $H_0$ : $\mu$ $\geq$ 21.1      {means that the mean Ohio score is above or equal the national average}

Alternate Hypothesis, $H_A$ : $\mu$ < 21.1      {means that the mean Ohio score is below the national average}

The test statistics that would be used here One-sample z test statistics as we know about population standard deviation;

                    T.S. =  $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n}}}$  ~ N(0,1)

where, $\bar X$ = sample mean Ohio score = 20.8

            $\sigma$ = population standard deviation = 3

            n = sample of Ohio = 500

So, test statistics  =  $\frac{20.8-21.1}{\frac{3}{\sqrt{500}}}$  

                              =  -2.24

The value of z test statistics is -2.24.

Now, at 0.1 significance level the z table gives critical value of -1.2816 for left-tailed test. Since our test statistics is less than the critical values of z as -2.24 < 1.2816, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the mean Ohio score is below the national average.