# Decide whether the normal sampling distribution can be used. If it can be used, test the claim about the population proportion p at the given level of significance α using the given sample statistics.

Decide whether the normal sampling distribution can be used. If it can be used, test the claim about the population proportion p at the given level of significance α using the given sample statistics.

Claim: p0.46; α = 0.01; Sample statistics: =0.38, n = 175

Solution

To use the normal sampling distribution,  both np and np must be greater then or equal to 5.

Begin by calculating np. The claim p 0.46 implies the hypothesized proportion p = 0.46.

np = 175(0.46)

= 80.5

Next calculate np. First determine the value for q.

q  = 1 –p

= 0.54

Use the value of q to find nq.

nq = 175(0.54)

=94.5

Since both np and nq are greater than or equal to 5, the normal sampling distribution can be used.

A null hypothesis is a statistical  hypothesis that contains a statement of equality. The alternative hypothesis is a complement of the null hypothesis. It is a statement that must be true if the null hypothesis is false, and it contains a statement of strict inequality. If the normal sampling distribution cannot be used, then the test cannot be performed.

The null and alternative hypotheses are given below.

0.46

Determine the critical value the using the fact that the test is a right-tailed test the level of significance is α = 0.01. Find the critical value, rounding to two decimal places.

The critical value is 2.33.

Calculate the standardized test statistic z using the following formula.

, where = p and =

Substitute the given value into the formula and evaluate, rounding to two decimal places.

=

= -2.12

The rejection region is shown to the right. If the standardized test statistic falls in the rejection region, then reject the null hypothesis. Otherwise, do not reject the null hypothesis.