# 1 Confidence intervals and hypothesis testing

## 1.1 Chronic illness (CI proportion)

In 2013, a Research Foundation reported that “45% of U.S. adults report that they live with one or more chronic conditions”. However, this value was based on a sample, so it may not be a perfect estimate for the population parameter of interest on its own. The study reported a standard error of about 1.2%, and a normal model may reasonably be used in this setting. Create a 95% confidence interval for the proportion of U.S. adults who live with one or more chronic conditions. Also interpret the confidence interval in the context of the study.

## 1.2 Website registration (CI proportion)

A website is trying to increase registration for first-time visitors, exposing 1% of these visitors to a new site design. Of 752 randomly sampled visitors over a month who saw the new design, 64 registered.

1. Check any conditions required for constructing a confidence interval.
2. Compute the standard error.
3. Construct and interpret a 90% confidence interval for the fraction of first-time visitors of the site who would register under the new design (assuming stable behaviors by new visitors over time).

## 1.3 Minimum wage (HT proportion)

Do a majority of US adults believe raising the minimum wage will help the economy, or is there a majority who do not believe this? A survey of 1,000 US adults found that 42% believe it will help the economy. Conduct an appropriate hypothesis test to help answer the research question.

## 1.4 Offshore drilling (HT difference proportions)

Results of a poll evaluating support for drilling for oil and natural gas off the coast of California are below.

Support 154 132
Oppose 180 126
Do not know 104 131
Total 438 389
1. What percent of college graduates and what percent of the non-college graduates in this sample support drilling for oil and natural gas off the Coast of California?
2. Conduct a hypothesis test to determine if the data provide strong evidence that the proportion of college graduates who support off-shore drilling in California is different than that of non-college graduates.

## 1.5 Sleep habits of New Yorkers (HI means)

New York is known as “the city that never sleeps”. A random sample of 25 New Yorkers were asked how much sleep they get per night. Statistical summaries of these data are shown below. The point estimate suggests New Yorkers sleep less than 8 hours a night on average. Is the result statistically significant?

n $$\bar x$$ s min max
25 7.73 0.77 6.17 9.78
1. Write the hypotheses in symbols and in words.
2. Check conditions, then calculate the test statistic, T, and the associated degrees of freedom.
3. Find and interpret the p-value in this context.
4. What is the conclusion of the hypothesis test?

## 1.6 Play the piano (CI and HI means)

Georgianna claims that in a small city renowned for its music school, the average child takes less than 5 years of piano lessons. We have a random sample of 20 children from the city, with a mean of 4.6 years of piano lessons and a standard deviation of 2.2 years.

1. Evaluate Georgianna’s claim (or that the opposite might be true) using a hypothesis test.
2. Construct a 95% confidence interval for the number of years students in this city take piano lessons, and interpret it in context of the data.
3. Do your results from the hypothesis test and the confidence interval agree? Explain your reasoning.

## 1.7 Car insurance savings (CI means, going backwards)

A market researcher wants to evaluate car insurance savings at a competing company. Based on past studies he is assuming that the standard deviation of savings is USD 100. He wants to collect data such that he can get a margin of error of no more than USD 10 at a 95% confidence level. How large of a sample should he collect?

## 1.8 Gaming and distracted eating (HT means difference)

A group of researchers are interested in the possible effects of distracting stimuli during eating, such as an increase or decrease in the amount of food consumption. To test this hypothesis, they monitored food intake for a group of 44 patients who were randomized into two equal groups. The treatment group ate lunch while playing solitaire, and the control group ate lunch without any added distractions. Patients in the treatment group ate 52.1 grams of biscuits, with a standard deviation of 45.1 grams, and patients in the control group ate 27.1 grams of biscuits, with a standard deviation of 26.4 grams. Do these data provide convincing evidence that the average food intake (measured in amount of biscuits consumed) is different for the patients in the treatment group? Assume that conditions for inference are satisfied.