# Sample size & Normal Probability

1. Construct a confidence interval for µd the mean differences d for population of paired data. Assume that the population of paired differences is normally distributed.

10 different families are tested for the number of gallons of water a day they use before and after viewing a conservation video. Construct a 90% confidence interval for the mean of differences.

I came up with 1.8 < µd < 7.8

2. Assume that the weight loss for the first month of a diet program varies between 6 and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost.

Less than 10 pounds.

I came up with 1/3

3. Find the d to the nearest tenth for the two sets of independent data.

I came up with 0.5

4. Assume Z is a standard normal variable, find the probability.

The probability that Z lies between -1.10 and -0.36.

I got 0.2238

5. Construct the indicated confidence interval for the differences between populations p1 - p2. Assume that the samples are independent and they have been randomly selected.

In a random sample of 500 people aged 20-24, 22% were smokers. In a random sample of 450 people aged 25-29, 14% were smokers. Construct a 95% confidence interval for the difference between the population proportion p1 - p2

I got 0.032 < p1 - p2 < 0.128

6. A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. Find P60, the score which separates the lower 60% from the top 40%.

I got 212.5

7. In a vote on the Clean Water bill, 41% of 205 Democrats voted for the bill while 40% of the 230 Republicans voted for it.

I got 0.212

8. Find the Margin of error for the 95% confidence interval used to estimate the population proportion.

N = 165, x = 96

I got 0.075264 or 0.0753 rounded

9. Find the minimum sample size you should use to assure that your estimate of p will be within the required margin of error and around the population p.

Margin of error 0.04; confidence interval 99%; from a prior study, p is estimated by 0.13.

10. Use the confidence level and sample data to find the margin of error E.

College students' annual earnings: 99% confidence; n = 74, x = $3967, σ = $874.

I got 261.7 rounded to $262

11. Use the given degree of confidence and sample data to construct a confidence interval for the population mean µ. Assume that the population has a normal distribution.

A sociologist develops a test to measure attitudes about public transportation. 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4. Construct the 95% confidence interval for the mean score of all such subjects.

I came up with 67.7 < µ < 84.7

12. Identify the null hypothesis H0 and the alternative hypothesis H1.

A researcher claims that 62% of voters favor gun control.

Duh, H0: p = 0.62; H1: p ≠ 0.62

13. Use the given information to find the p-value.

The test statistic in a right-tailed test is z = 1.43.

I got 0.0764

14. Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, student t distribution, or neither.

Claim: µ = 111. Sample data: n = 10, x = 101, s = 15.3. The sample data appear to come from a normally distributed population with unknown µ and σ.

I chose student t distribution since the standard deviation is unknown.

15. Assume that a hypothesis test of the given claim will be conducted. Identify the type I error for the test.

Carter Motor Company claims that its new sedan, the Libra, will average better than 30 miles per gallon in the city. Identify the type I error for the test.

I found the error of rejecting the hypothesis that the mean is 30 mpg when it really is 30 mpg.

16. Assume the data has a normal distribution and the number of observations is greater than fifty. Find the critical z value used to test the null hypothesis.

α = 0.1 for a two-tailed test.

I got + 1.645

17. Find the number of successes, x, suggested by the given statement.

A computer manufacturer randomly selects 2360 of its computers for quality assurance and finds that 2.54% of these computers are found to be defective.

I have 59.944 rounded to 60.

18. Assume in one region, the September energy consumption levels for single-family homes are found to be normally distributed with a mean of 1050 KWh (kilo watt per hour) and standard deviation of 218 KWh.

If 50 different homes are selected, find the probability that their mean energy consumption level for September is greater than 1075 KWh.

I came up with 0.2090

19. A final exam has a mean of 73 with a standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is less than 70.

I got 0.0301

20. Find the critical value za/2 that corresponds to a degree of confidence of 98%.

I got 2.33

https://brainmass.com/statistics/hypothesis-testing/sample-size-normal-probability-396520

#### Solution Preview

1. Construct a confidence interval for µd the mean differences d for population of paired data. Assume that the population of paired differences is normally distributed.

10 different families are tested for the number of gallons of water a day they use before and after viewing a conservation video. Construct a 90% confidence interval for the mean of differences.

I came up with 1.8 < µd < 7.8 Correct answer

2. Assume that the weight loss for the first month of a diet program varies between 6 and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost.

Less than 10 pounds.

I came up with 1/3 Wrong answer

Correct answer

Let X be the weight loss for the first month of the diet program. Then X follows uniform distribution and the pdf of X is given by,

We need P (X < 10).

P (X < 10) = = 4/6 = 2/3

3. Find the d to the nearest tenth for the two sets of independent data.

I came up with 0.5 Correct answer

4. Assume Z is a standard normal variable, find the probability.

The probability that Z lies between -1.10 and -0.36.

I got 0.2238 Correct answer

5. Construct the indicated confidence interval for the differences between populations p1 - p2. Assume that the samples are independent and they have been randomly selected.

In a random sample of 500 people aged 20-24, 22% were smokers. In a random sample of 450 people aged 25-29, 14% were smokers. Construct a 95% confidence interval for the difference between the population proportion p1 - p2

I got 0.032 < p1 - p2 < 0.128 ...

#### Solution Summary

The solution provides step by step method for the calculation of probability using the Z score and sample size. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.