# Confidence Interval

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Problem 1: A bank in a small town has 10,000 customers. A national survey on the banking habits of people in U.S. shows that 80% of the people with income higher than 50,000 dollars have both savings and checking accounts and also shows that the average number of banking operations that a person aged 18 and over performs per week is 5. The manager of the bank decides to do a survey among the customers of his bank and takes a simple random sample of 200 customers aged 18 and over. In the sample, the average number of banking transactions per week is 8 with standard deviation equal to 10.

a) The average number of times a customer carries out banking transactions per week is _________ give or take ________ or so. Show how you computed your answers.

b) Give a 90% confidence interval for the average number of banking operations per week for the town residents aged 18 and over. Show your working.

c) Is the apparent difference in banking habits between the nation and the customers of the bank real or just due to chance?

d) A 95% confidence interval gives a range of values for the _______which are plausible according to the observed data. Fill in the blanks.

(Possible answer: (A) Population average, (B) Sample average).

e) The sample standard deviation measures how far _______ is from sample average.

The standard deviation for the sample average measures how far __________is from the population average - for typical __________.

To fill in the blanks, choose among: (A) number of bank operations, (B) average number of operations, (C) samples, (D) customers aged 18 and over, (E) bank, (F) person with high income.

Problem 2: In a e-commerce site, web applications that handle purchases or queries should perform tasks in a timely matter at any time. In particular web sites should be able to handle high-demand situations with many users accessing the site at the same time. No one wants to lose customers!

You suspect that in cases of high traffic, there is a 30% chance of waiting longer than 8 seconds to download a certain page. You run a usability test to evaluate your idea. A certain task consisting in accessing a certain page is performed 40 times. The test results are that in 15 out of 40 times the wait was longer than 8 seconds.

The sample size is _____

The sample count is _____ The sample proportion is __________

The expected value for the sample proportion is _______

The standard deviation of the sample proportion = _________

Are the results of your usability test consistent with your guess of 30% chance?

Problem 3 : A 95% confidence interval for p is (0.3, 0.6). Based on the data from which the confidence interval was constructed, someone wants to test H0: p=0.5 versus Ha: p not equal to 0.5. The P-value will be:

a) greater than 0.05

b) less than 0.05

c) equal to 0.05

why is this your answer?

For which null hypothesis will p=0.05?

a) H0: p=0.3

b) H0: p<=0.3

c) H0: p>=0.3

why is this your answer?

Problem 4 : A scientist computes a 90% C.I. for a population average m to be (4.38, 6.02). Using the same data, she also computes a 95% C.I. to be (4.22, 6.18), and a 99% C.I. to be (3.91, 6.49). Now she wants to test H0: m = 4 versus Ha: m not equal to 4. With regard to the P-value, which one of the following statements is true? Show your working.

a) P-value > 0.10

b) 0.05 < p-value < 0.10

c) 0.01 < p-value < 0.05

d) P-value < 0.01.

https://brainmass.com/statistics/confidence-interval/confidence-interval-pvalue-hypothesis-testing-213760

#### Solution Preview

See the attached file.

Problem 1

A bank in a small town has 10,000 customers. A national survey on the banking habits of people in U.S. shows that 80% of the people with income higher than 50,000 dollars have both savings and checking accounts and also shows that the average number of banking operations that a person aged 18 and over performs per week is 5. The manager of the bank decides to do a survey among the customers of his bank and takes a simple random sample of 200 customers aged 18 and over. In the sample, the average number of banking transactions per week is 8 with standard deviation equal to 10.

Sample mean= 8

Sample Standard deviation = 10

Sample size= n= 200

Standard error of mean = standard deviation / √n = 10/√200 = 0.7071

99.7% of the values will be between ±3 standard error of mean from the sample mean

90% confidence interval is ± 1.6449 standard error of mean from the sample mean

Sample mean ±3 standard error of mean= 8 ±3 x 0.7071= 8±2.12

Sample mean ±1.6449 standard error of mean= 8 ±1.6449 x 0.7071= 8±1.16

a) The average number of times a customer carries out banking transactions per week is _8________ give or take _2_______ or so. Show how you computed your answers.

Sample mean= 8

Sample Standard deviation = 10

Sample size= n= 200

Standard error of mean = standard deviation / √n = 10/√200 = 0.7071

99.7% of the values will be between ±3 standard error of mean from the sample mean

Sample mean ±3 standard error of mean= 8 ±3 x 0.7071= 8±2.12

Lower Limit= 8+2.12 = 10.12

Lower Limit=8-2.12= ...

#### Solution Summary

The solution discusses questions related to the confidence interval, p-value and hypothesis testing.

Confidence Intervals, Samples, and Unknown Variance

Let (X1, X2,..., Xn) be a sample, where each Xi is a random variable of normal distribution with mean mu and variance sigma². Let us suppose that n = 20, and sigma² = 9. An experiment has yielded the results (X1, X2,..., X20), and we have calculated that the empirical mean x20 = 2.09.

1) Give a confidence interval with level of confidence 90% for mu.

2) How big would the sample have to be for the interval to be half as long?

3) Let us now suppose the variance is not known. Knowing that counting from i=1 to 20 (xi - x20)² = 14.6, give a confidence interval with level of confidence 90% for the value of mu.

4) We now suppose we know mu is known and is equal to 2. Give a confidence interval with a confidence level of 90% for the value of sigma².

5) Same question if we do not know the value of mu.

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