Question

In a recent Pew Research Center poll, 1503 adults were randomly selected and asked this question: "On average, how many hours of sleep do you get in a 24-hour period?" 55% answered 7 to 9 hours.

1) Explain what you could do statistically with this data. For example, could you build a confidence interval, perform a hypothesis test, use the normal distribution to find a probability, organize the information into graphs using descriptive statistics, etc. Explain at least two things that you could do with the data and how it would be useful or helpful in interpreting the data.

2) A common criticism of surveys is that they poll only a very small percentage of the population and therefore cannot be accurate. Is the sample that you choose too small? Explain why or why not.

Answer 1

1)

Given data is sample proportion of adults who answered that number of hours of sleep they get is 7 to 9.

We can perform various statistical operations with the sample proportion. We can build a confidence interval, perform a hypothesis test, use the normal distribution to find a probability and organize the information into graphs using descriptive statistics.

Building a 95% confidence interval -

Standard error of sample proportion, se = = 0.01283

Z value for 95% confidence interval is 1.96.

Margin of error = z * se = 1.96 * 0.01283 = 0.0251

95% confidence interval is,

(0.55 - 0.0251, 0.55 + 0.0251)

(0.5249,  0.5751)

Interpretation - We're 95% confident that the interval (0.5249,  0.5751) captured the true proportion of who get 7 to 9 hours of sleep.

Perform a hypothesis test -

Null Hypothesis H0: True proportion of who get 7 to 9 hours of sleep is 0.5. That is p = 0.5

Alternative Hypothesis Ha: True proportion of who get 7 to 9 hours of sleep is not 0.5. That is p 0.5

Test statistic, z = (0.55 - 0.5) / 0.01283 = 3.90

Since the test statistic is greater than 1.96 (critical z value for 5% significance level), we reject null hypothesis H0 and conclude that there is significant evidence that True proportion of who get 7 to 9 hours of sleep is not 0.5.

2)

n = 1503

p = 0.55

np = 1503 * 0.55 = 826.65

n(1-p) = 1503 * (1 - 0.55) = 676.35

Since both np and n(1-p) are greater than 10, we can assume that sample is too big to be accurate.