Question

1- The following data show the number of times a sample of students cut their hair in a year:

7,9,20,18,7,10,12,12,20,18,6,8,9,12,20,18,16,12,6,7,10,12,4,6,9

Obtain the 95% confidence interval for the mean number of times the students cut their hair in a year.

2- A survey of 1006 Americans showed that 804 believe that Congress should consider reallocating federal subsidies from fossil fuels to solar. Find the 94% confidence interval for the proportion of Americans who believe that Congress should so reallocate subsidies. Obtain the 90% confidence interval for the population proportion.

3- A survey of 1006 Americans showed that 804 believe that Congress should consider reallocating federal subsidies from fossil fuels to solar. Find the 90% confidence interval for the proportion of Americans who believe that Congress should so reallocate subsidies. Obtain the standard error for the sample proportion.

4- The number of calories in a hamburger depends on what ingredients are included. The following are determinations of calorific values of a sample of homemade large hamburgers:

350 375 400 356 378 379 340 320 330 324 360 390

Assume that the calorific values for the hamburgers are normally distributed.

a) Find the standard error of the sample mean.

b) Obtain the 95% confidence intervals for the mean calorific values of the burgers.

5- The following show the number of hours a sample of High school students from a local high school studied a week:

21, 15, 18, 24, 15, 18, 20, 23, 24, 12, 10, 20, 12, 18, 20, 14, 17

Assume that the weekly study hours are normally distributed.

a) find the 90% confidence intervals for the mean number of study hours.

b) find the margin of error for the 95% confidence interval of the mean number of study hours for all students in this high school. The margin of error equals half length of the confidence interval.

6- In a study, 566 patients were treated with a medication for a disease of which 202 recovered fully. In the same study, 365 patients were given placebo (an inert substance that looks like the medication), 96 recovered fully.

a) Obtain a 95% confidence interval for the difference between proportions of patients who recovered fully for both groups.

b) Test whether there is difference between proportions of patients who recovered fully for both groups. Use α=0.05

7- A survey of 1006 Americans showed that 804 believe that Congress should consider reallocating federal subsidies from fossil fuels to solar. Find the 90% confidence interval for the proportion of Americans who believe that Congress should so reallocate subsidies. Obtain the standard error for the sample proportion.

Answer 1

Let us answer question 1

The number of times a sample of students cut their hair in a year is as follows

7,9,20,18,7,10,12,12,20,18,6,8,9,12,20,18,16,12,6,7,10,12,4,6,9

We have to find the 95% confidence interval for the mean number of times the students cut their hair in a year.

For this, first we find the mean of the values

The standard devaition of the sample is given by,

At 95% confidence level, critical value of z is given by, z*=1.96

The 95% confidence interval is given by,

  

Therefore, the required 95% confidence interval of mean number of times the students cut their hair in a year is