Question

For the following information, determine whether a normal sampling distribution can be used, wherep is the population proportion, a is the level of significance, p is the sample proportion, and n is thesample size. If it can be used, test the claim.Claim: p20.28; a=0.08. Sample statistics: P=0.20, n = 160O A.Zo =(Round to two decimal places as needed. Use a comma to separate answers asneeded.)OB. A normal sampling distribution cannot be used.If a normal sampling distribution can be used, identify the rejection region(s). Select the correctchoice below and, if necessary, fill in the answer box(es) to complete your choice.<2

Answer 1

We have an hypotesis test where the claim is that π < 0.28.

(NOTE: we will use π for the population proportion and p for the sample proportion).

The sign in the claim, as being "<", shows that we have a one-tail test.

The level of significance is 0.08 and the sample statistics are p = 0.20 and n = 160.

For test of proportions and with sample sizes greater than 30, a normal sampling distribution can be used.

We will start by calculating the critical value for z. The z-value is a normalization of the distribution in order to use the probabilities in the standard normal distribution.

NOTE: The z-value for a proportion can be expressed as the difference between the sample proportion and the population proportion (expressed in the null hypothesis) divided by the standard error.

In this case, we only need to know the critical value of Z for this and this depends only of the significance level and the type of test.

We know, from the significance level of 0.08, that the probability that the z-statistic is greater than the the critical z0 by pure chance (Type I error) has to be 0.08.

Then, we can write:

`P(zFrom the standard normal distribution we get:<p></p>[tex]P(z<-1.40507)=0.08`

Then, the value of the critical z0 is -1.41, rounded to 2 decimals.

This means that if the z-statistic is smaller than z0 = -1.41, the null hypothesis is rejected.

In this case, the claim is that the population proportion is smaller than 0.28. This should be stated as the alternative hypothesis, as it is what it is trying to be proved with statistically significant evidence (NOTE: alternative hypothesis don't use equal signs).

On the other hand, the null hypothesis will state the opposite: that the population proportion is equal or greater than 0.28.

The rejection region, as we only have one tail, will be the z < -1.41.

If z is smaller than -1.41, it means that the sample statistic is significantly smaller (from the statistics point of view) than the population proportion stated in the null hypothesis, so this can be rejected.