Part 1 : Show the necessary steps

You are doing marketing research about yogurt consumption.Based on your previous research, you feel sure that the population standard deviation of the number of yogurts consumed per year is 6.In other words, is known and = 6.You take a sample of 50 consumers and the sample mean is 32 yogurts consumed per year.In other words, x= 32.You want to develop a confidence interval for (the population mean) such that you are 90% confident that the true value of lies within the interval.

1.What is the value of ?

2.What is the value of /2 ?

3.What is the value of z/2?

4.What is the value of the margin of error (MOE)?

5.State the confidence interval showing your confidence level that lies between two numbers.

6.Redo the problem for a 95% confidence interval.

7.Redo the problem for a 99% confidence interval.

8.What happens to the margin of error and the confidence interval as the confidence level increases?

You took over William Gosset's old job, and you are measuring the quantity of a certain enzyme found in Guiness beer.You are pretty sure that the distribution of this enzyme in bottles of beer has a normal distribution.You take a small sample of bottles of Guiness and find the following values for the quantity of the enzyme:1081215131165.

You want to determine a range of values for the mean quantity of this enzyme and you want to be 95% confident that your range contains the true population mean.

9.What is your point estimate for the population mean quantity of this enzyme in Guiness?

10. What is your point estimate for the population standard deviation of the quantity of this enzyme in Guiness?

11. What is the value of ?

12. How many degrees of freedom are there?

13. What is the appropriate value for t_/2,df ?

14. What is the margin of error?

15. What is your confidence interval?Be sure to show it in a form that shows your level of confidence that lies between two numbers.

Part 2: Show the necessary steps

A certain small town has a population of 5000 residents.You want to calculate a confidence interval for the average number of gallons of gas bought per month by the residents of this town.You want to be 95% confident that the true value of the population mean is within your interval, and you need to have a margin of error no higher than 10 gallons per month.Based on your previous research in similar towns, you believe that the population standard deviation is 50 gallons per month.

Give the appropriate statistical symbol or formula for each of the following numbers in this problem:

1.5000 =

2.50 =

3.95% =

4.What is the value of ?

5.What is the value of /2 ?

6.What is the value of z_/2?

7.What will be the width of your confidence interval from the smaller number to the larger number?(Hint:Think about how the MOE is related to the width of the confidence interval.)

8.Which of the following statements is the correct way to make a general statement about the confidence interval?

a. We are 95% confident that xis in the interval to -10 to +10

b. We are 95% confident that xis in the interval to - 10 to + 10

c. We are 95% confident that is in the interval to x - 10 to x + 10

d. We are 95% confident that is in the interval to x - 50 to x + 50

e. We are 2.5% confident that is in the interval to x - 10 to x + 10

f.We are 5% confident that is in the interval to x - 10 to x + 10

9.How many residents of the town do you need to include in your sample in order to develop a confidence interval of the appropriate width and the appropriate confidence level?

10. You also need to create confidence interval for gasoline consumption for a big city with 1,000,000 residents.This confidence interval must have a width of 20 and a 95% confidence level.The estimated population standard deviation in the big city is also 50 gallons per month.How many residents of the big city do you need to sample in order to create confidence interval?

You are working on a political campaign to pass a proposition in California.You take a simple random sample of 400 California voters and 100 say they plan to vote yes on your proposition.Use four decimal points in all of your answers.

11. What is your point estimate for the population proportion that plans to vote yes?

12. What is your estimate of the standard error of the proportion, ?Round to 4 decimals.

13. What is the margin of error in your sample if the confidence level is 95%?Round to 4 decimals.

14. Calculate the 95% confidence interval for the population proportion based on your sample.Be sure to state it in a complete sentence that shows your confidence level that the population proportion lies between two values.

15. The leader of your campaign does not think your results are accurate enough.She wants you to take a new sample, and this time she wants you to be 95% sure that the sample proportion is within .03 of the actual population proportion of California voters who plan to vote for the proposition. Based on your previous sample, you now have a planning value for p.Use this value as p* and calculate how many California voters you need to include in your new sample in order to meet the campaign leader's requirement.