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 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 of significance α using the given sample statistics.

Claim p 0.31; α = 0.05; Sample statistics : : = 0.26, n = 210

 

Solution

 

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

Begin by calculating np. The claim p < 0.31 implies the hypothesized proportion p = 0.31

np  = 210 (0.31)

= 65.1

Next calculate np. First determine the value for q.

q = 1 – p

= 0.69

Use the value of q to find np.

np  = 210(0.69)

= 144.9

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

A  null hypothesis is a statistical hypothesis that contain 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 contain a statement of strict 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.

:p = 0.31

Determine the critical values using the fact that the test is a two-tailed test and the level of significance is α = 0.05. Find the critical values using technology, rounding to two decimal places.

The  critical values are – 1.96 and 1.96.

Calculate the standardized test statistic z using the following formula.

, where = p and =

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

=

= – 1.57

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.