Question

AT&T would like to test the hypothesis that the proportion of 18- to 34-year-old Americans that own a cell phone is less than the proportion of 35- to 49-year-old Americans. A random sample of 200 18- to 34-year-old Americans found that 126 owned a smartphone. A random sample of 175 35- to 49-year-old Americans found that 119 owned a smartphone. If Population 1 is defined as 18-to 34-year-old Americans and Population 2 is defined as 35- to 49-year-old Americans, and using a = 0.01, the conclusion for this hypothesis test would be that because the absolute value of the test statistic is less than the absolute value of the critical value, AT&T can conclude that the proportion of 18- to 34- year-old Americans that own a cell phone is less than the proportion of 35- to 49-year-old Americans less than the absolute value of the critical value, AT&T cannot conclude that the proportion of 18-to 34- year-old Americans that own a cell phone is less than the proportion of 35- to 49-year-old Americans more than the absolute value of the critical value, AT&T can conclude that the proportion of 18- to 34- year-old Americans that own a cell phone is less than the proportion of 35- to 49-year-old Americans more than the absolute value of the critical value, AT&T cannot conclude that the proportion of 18-to 34- year-old Americans that own a cell phone is less than the proportion of 35- to 49-year-old Americans

Answer 1

The pooled proportion here is computed as:
P = (126 + 119) / (200 + 175) = 0.6533

The standard error here is computed as:

$SE = \sqrt{P(1-P)(\frac{1}{n_1} + \frac{1}{n_2})} =\sqrt{0.6533( 1 - 0.6533)(\frac{1}{200}+ \frac{1}{175})} = 0.0493$

The sample proportions are computed here as:
p1 = 126/200 = 0.63

p2 = 119/175 = 0.68

As we are testing here whether p1 < p2, the test statistic here is computed as:

$z^* = \frac{p_1 - p_2}{SE}= \frac{0.63 - 0.68}{ 0.0493 } = -1.0142$

For 0.01 level of significance, as this is a one tailed test, we have from the standard normal tables here:
P(Z < -2.326) = 0.01

Therefore -2.326 is the critical value here.

As the absolute value of the test statistic here is less than the absolute value of the critical value, therefore the test is not significant here and we cannot conclude that the proportion of 18 to 34 year ods Americans that own a cell phone is less than the proportion of 35 to 49 year old Americans.