1. A survey is conducted asking US adults if they believe inmates in prisons in NY State convicted of non-violent crimes should be released early if they have less 6 months left on their sentence to alleviate the spread of the Coronavirus. In a random sample of 300 US adults, 210 said they believe these inmates should be released early. Construct a 90% confidence interval for the population proportion.
a) What is your first step and why?
b) Calculate the correct answer. Show ALL your work.
2. The MTA wants to determine the average number of homeless people riding the trains between 12am and 5am. They want a 95% confidence interval with a margin of error of 1.5 people. If the data is normally distributed, and the population standard deviation is 9.8 people, find the minimum number of nights that they should count the number of homeless people riding the trains between the hours of 12am and 5am.
a) Are you using a t-distribution or a normal distribution? Explain how you know.
b) Calculate the correct answer. Show ALL your work.
3. In a random sample of 40 US household with 3 people, the mean amount of money spent on food per week is $620 and the standard deviation is $52. Construct a 99% confidence interval for the mean amount of money spent on food per week for US households with 3 people.
a) Calculate the correct answer. Show ALL your work.
b) What would you do differently if you were told the population standard deviation is $52 instead?
4. Suppose a clinical study is trying to determine the mean number of years added to a person’s life after receiving a particular drug that treats Parkinson’s disease. In a random sample of 28 patients, the mean number of years added to their life was 4.6 years. From past studies the population standard deviation was found to be 1.9 years. If the population is normally distributed, construct a 95% confidence interval for the population mean age.
a) What are two things this student did wrong and why were they wrong? Do NOT just write the correct answer.
The degrees of freedom = n – 1 = 27 c = .95
tc = 2.052
= 2.052 ∗ 1.9 = .75 √28
4.6 + .75 = 5.35 3.85 < < 5.35
4.6 - .75 = 3.85
b) Write the correct answer. Show ALL your work.
5. You want to estimate with 80% confidence the number of defendants who request a public defender. Your estimate must be accurate within 4% of the population proportion. Find the minimum sample size needed using a prior survey that found 42% of defendants request a public defender.
a) Calculate the correct answer. Show ALL your work.
b) Without calculating an answer, do you think your minimum sample size would be larger or smaller if your estimate had to be accurate within 1% of the population proportion with 80% confidence? EXPLAIN your answer.
