Question

The report "Digital Democracy Survey"† describes a large national survey. In a representative sample of Americans ages 14 to 18 years, 45% indicated that they usually use social media while watching TV. Suppose that the sample size was 500. (a) Is there convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV? Use a significance level of 0.05. State the appropriate null and alternative hypotheses. H0: p = 0.5 versus Ha: p ≠ 0.5 H0: p < 0.5 versus Ha: p > 0.5 H0: p ≠ 0.5 versus Ha: p = 0.5 H0: p = 0.5 versus Ha: p > 0.5 H0: p = 0.5 versus Ha: p < 0.5 Find the test statistic and P-value. (Use a table or technology. Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = State the conclusion in the problem context. Fail to reject H0. We do not have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV. Reject H0. We have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV. Fail to reject H0. We have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV. Reject H0. We do not have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV. (b) Suppose that the sample size had been 100 rather than 500 and that 45% of those in the sample indicated that they usually use social media while watching TV. Based on this sample of 100, is there convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV? Use a significance level of 0.05. State the appropriate null and alternative hypotheses. H0: p = 0.5 versus Ha: p > 0.5 H0: p = 0.5 versus Ha: p < 0.5 H0: p ≠ 0.5 versus Ha: p = 0.5 H0: p = 0.5 versus Ha: p ≠ 0.5 H0: p < 0.5 versus Ha: p > 0.5 Find the test statistic and P-value. (Use a table or technology. Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = State the conclusion in the problem context. Fail to reject H0. We do not have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV. Reject H0. We have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV. Fail to reject H0. We have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV. Reject H0. We do not have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV. (c) Explain why different conclusions were reached in the hypothesis tests of parts (a) and (b). Getting out of 500 people responding this way as opposed to out of 100 provides much evidence that fewer than half of Americans ages 14 to 18 years usually use social media while watching TV.

Answer 1

a ) To test :

Test statistic :

where ,

The value of the test statistic :  

Now ,

Reject H₀. We have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV.

b ) To test :

Test statistic :

where ,

The value of the test statistic :  

Now ,

Fail to reject H₀. We do not have convincing evidence that less than half of Americans ages 14 to 18 years usually use social media while watching TV.

c ) Getting out of 500 people responding this way as opposed to out of 100 provides much evidence that fewer than half of Americans ages 14 to 18 years usually use social media while watching TV.