Although seniors consistently have lower rates of technology adoption than the g

Question

Although seniors consistently have lower rates of technology adoption than the general public, some seniors are more digitally connected than ever. Today, 41% of all Americans 70-74 years old say they use social networking sites like Facebook or Twitter. Suppose this is the correct rate for the 70-74 year old population of Americans. Researchers would like to assess if this rate appears to decline with age; specifically, if this rate is lower for U.S. adults aged 80 and older. A random sample of 12 U.S. adults aged 80 and older was taken, and 2 of the sampled adults reported using social networking sites like Facebook or Twitter. A 10% significance level is to be used for this analysis.

The hypotheses to be tested are H0: p = 0.41 versus Ha: p < 0.41.

Think about:
(i) Who are the individuals that we are trying to learn about in this study? (i.e., define the population)
(ii) What characteristic are we measuring for in this population? (i.e., define the "success")

Using (i) and (ii), write 2-3 sentences to explain the research question behind the hypotheses to be tested, that is, to clearly explain in words (without notation) the meaning of the alternative hypothesis Ha: p < 0.41.

If the null hypothesis is true, what is the distribution of the test statistic (that is, which distribution should be used to find the p-value)?

(i) Report the observed value of the test statistic for testing the hypotheses in part (a). Be sure to provide both correct notation and the value.
(ii) Compute the p-value for this test (it is always expected that you show all work).

Explanation / Answer

Here, our sample consists of U.S adults aged 80 and above and we are studying about the proportion of these people using social networking sites. If a person uses social networking sites, we call it ‘success’ and else ‘failure’.
The alternative hypothesis is the hypothesis which we are trying to prove true, i.e the proportion of U.S adults aged 80 and above using social networking sites is less than 41%.

If the null hypothesis is true, the test statistic follows binomial distribution.

> binom.test(2,12,p=0.41,alt="l") #Rcode

Exact binomial test

data: 2 and 12

number of successes = 2, number of trials = 12, p-value = 0.07332

alternative hypothesis: true probability of success is less than 0.41

95 percent confidence interval:

0.0000000 0.4381054

sample estimates:

probability of success

             0.1666667

Since p-value < 0.1, we reject the null hypothesis at 10% level and conclude that the proportion of U.S adults aged 80 and above using social networking sites is significantly lower than 41%.

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