Part 1) Amount of Sleep(7 points)

A sleep study administered to U.S. adults showed that the amount of sleep (in hours) they get in a 24-hour period is normally distributed with a mean of 6.5 hours and a standard deviation of 1.25 hours. Use normal probability calculations to answer the following questions.

Show your calculator functions or Excel commands to receive full credit. Round your answers to 4 decimals.

A. How many hours of sleep did you get last night (round to the nearest quarter of an hour)?

____________

B. Ask an adult friend or a family member how many hours of sleep he/she got last night (round to the nearest quarter of an hour). Report below. Make sure it is different than your sleep amount.

____________

1. (1 pt) What is the probability that a randomly selected U.S. adult slept less than your friend or family member did last night?

1. (1 pt) What is the probability that a randomly selected U.S. adult slept more than you did last night?

1. (1 pt) Doctors recommend 8 hours of sleep per day for adults to have the health benefits of sleep. What percent of U.S. adults sleep less than this recommended amount?

1. (1 pt) A colleague at work says that she usually sleeps less than 4 hours each day. Is her sleep amount unusual? Justify your answer by calculating its probability.

1. (1 pt) 10% of U.S. adults sleep more than how many hours?

1. (1 pt) A simple random sample of 25 U.S. adults is taken. What is the mean and standard deviation of the sampling distribution of sample means? Report your answers using the proper symbols.

1. (1 pt) A simple random sample of 25 U.S. adults is taken. What is the probability that the sample mean hours slept is less than what you slept last night?

Part 2) Effect of Covid Lockdowns on Speech and Language Skills(6 points)

A research study in England⁽¹⁾ done by The Education Endowment Foundation (EEF) suggests the measures taken to combat the pandemic have deprived the youngest children of social contact and experiences essential for increasing vocabulary. The study sampled 58 primary schools and found 76% of schools stated students starting school in 2020 needed more communication support than in previous school years.

Show your work or calculator functions to answer the following questions.

1. (2 pt)Find the mean and standard deviation of the sampling distribution of the sample proportion of schools stating students starting school in 2020 needed more communication support than in previous years.

1. (2 pt)What is the probability that the sample proportion of schools stating students starting school in 2020 needed more communication support than in previous years is greater than 70%?

1. (2 pt)What is the probability that the sample proportion of schools stating students starting school in 2020 needed more communication support than in previous years is between 50% and 80%?

Part 3) Confidence Intervals and Sample Size for and p

A. Confidence Interval and Sample Size for -- How much do you read?(9 points)

A recent Gallup poll asked a random sample of 1006 Americans "During the past year about how many books, either hardcover or paperback, did you read either all or part of the way through?" The results of the survey gave a sample mean of 13.4 books and a standard deviation of 16.6 books read all or part of the way through.

Choose a number between 80 and 98. This is the level of confidence you will use for this section of problems. What is your number? _____________

1. (2 pts) With the confidence level you chose, construct that % confidence interval for the population mean number of books Americans read either all or part of the way through. What is the margin of error for this confidence interval?

1. (1 pt) Interpret the interval found above. Be sure to include the chosen confidence level and the context of the problem in the interpretation.

1. (1 pt)A book publishing company claims that an average of 14 books were read either all or part of the way through based on their publishing goals that were met.Does the interval found in #1 above contradict or support this claim?

1. (3 pts)Find the margin of error for the following confidence intervals and compare these to the margin of error for the confidence interval you found above in #1.
2. 75% confidence interval. Margin of error =
3. 99% confidence interval. Margin of error =
4. How do these margins of errors compare to the margin of error for the confidence interval found in #1 above?

1. (2 pts)What sample size is required if = 15.1 and a margin of error of 2 is desired with the confidence level chosen in #1 above? Be sure to show the formula used and work on solving the problem.

B. Confidence Interval and Sample Size for pEducation of Fourth Graders in the U.S.?(8 points)

A survey conducted by Conquest Communications Group randomly sampled 700 registered voters in the U.S. 258 of the 700 sampled voters correctly answered that fourth graders in the U.S. are above average for overall education compared to other countries.

Choose a number between 80 and 96. This is the level of confidence you will use for this section of problems. What is your number? _____________

1. (2 pts) With the confidence level you chose, construct that % confidence interval for the population proportion of voters that answered correctly that 4^th graders in the U.S. are above average in overall education compared to other countries. What is the margin of error for this confidence interval?

1. (1 pt) Interpret the interval found above. Be sure to include the chosen confidence level and the context of the problem in the interpretation.

1. (3 pts)Find the margin of error for the following confidence intervals and compare these to the margin of error for the confidence interval you found above in #1.
2. 79% confidence interval. Margin of error =
3. 98% confidence interval. Margin of error =
4. How do these margins of errors compare to the margin of error for the confidence interval found in #1 above?

1. (2 pts)What sample size is required if a margin of error of 4% is desired using the confidence level chosen in #1 above? Be sure to show the formula used and work solving the problem.