Question

In 2015, 45% of Americans had only a cell phone (and no landline). A research firm plans to investigate whether this percentage has changed in the meantime. In a recent sample of 1252 people, 613 reported having only a cell phone.

Fill in the blanks: H0:p=________ and Ha:p ? __________

Calculate p?

Are both np0 and (1-p0)at least 15? Show the calculations.

Draw a bell (normal) curve with the mean and two standard errors (in both directions) marked. Mark p? and shade the appropriate area.

Calculate the z-score. Show the calculation.

Use Table A to find the (two-tail) P-value.

The sample proportion (circle one: is / is not ) “close” (within 2 standard errors) to the claimed population proportion. Therefore we (circle one: will / will not ) reject the null hypothesis and conclude that there (circle one: is / is not ) evidence that the percentage of Americans who only have a cell phone is different from what it was in 2015.

Suppose that we were interested in testing the claim that more people today than in 2015 only have a cell phone. Using correct symbols, write the null and alternative hypotheses for this situation.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P = 0.45
Alternative hypothesis: P 0.45

Note that these hypotheses constitute a two-tailed test.  

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

S.D = sqrt[ P * ( 1 - P ) / n ]

S.D = 0.01406
z = (p - P) /S.D

z = 2.82

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than -2.82 or greater than 2.82.

Thus, the P-value = 0.005.

Interpret results. Since the P-value (0.005) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that the percentage of Americans who only have a cell phone is different from what it was in 2015.