Exam 8: Confidence Interval Estimation

The confidence interval obtained will always correctly estimate the population parameter.

A major department store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain's new store.Fifteen credit card accounts were randomly sampled and analysed with the following results: $\bar { X }$ = $50.50 and S = 20. -Assuming the distribution of the amount spent on their first visit is approximately normal,what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for $\mu$ ?

Instruction 8-14 After an extensive advertising campaign, the manager of a company wants to estimate the proportion of potential customers that recognise a new product. She samples 120 potential consumers and finds that 54 recognise this product. She uses this sample information to obtain a 95% confidence interval that goes from 0.36 to 0.54. -Referring to Instruction 8-14,it is possible that the true proportion of people that recognise the product is between 0.36 and 0.54.

Instruction 8-4 To become an actuary, it is necessary to pass a series of 10 exams, including the most important one, an exam in probability and statistics. An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program. They take a sample of 8 actuarial students in this program and determine that their scores are: 2, 5, 8, 8, 7, 6, 5, and 7. This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program. -Referring to Instruction 8-4,it is possible that the confidence interval obtained will not contain the mean score for all actuarial students in the special class.

The confidence interval estimate of the population mean is constructed around the sample mean.

Private colleges rely on money contributed by individuals for much of their major building expenses.Much of this money is put into a fund called a trust,and the college spends only the interest earned by the fund.A recent survey of 8 private colleges in Australia revealed the following trusts (in millions of dollars): 60.2,47.0,235.1,490.0,122.6,177.5,95.4,and 220.0.What value will be used as the point estimate for the mean trust of all private colleges in Australia?

A random sample of 50 provides a sample mean of 31 with a standard deviation of S =14.The upper bound of a 90% confidence interval estimate of the population mean is 34.32.

Instruction 8-19 A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students. The results were 1,344 answered "YES" and 1,456 answered "NO". -Referring to Instruction 8-19,a 99% confidence interval for the proportion of student population who feel comfortable reporting cheating by their fellow students is from ________to ________.

Instruction 8-6 A sample of salary offers (in thousands of dollars) given to students majoring in management is: 28, 31, 26, 32, 27, 28, 27, 30, 31, and 29. Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3. -Referring to Instruction 8-6,95% of the salary offers are between 27.5 and 30.3.

Instruction 8-15 A wealthy real estate investor wants to decide whether it is a good investment to build a high-end shopping complex in a suburb of Sydney. His main concern is the total market value of the 3,605 houses in the suburb. From past experience, the standard deviation of market housing prices is estimated to be $81,000. He commissioned a statistical consulting group to take a sample of 200 houses and obtained a sample average market price of $450,000 and a sample standard deviation of $77,400. The consulting group also found out that the average differences between market prices and appraised prices was $250,000 with a standard deviation of $6,800. Also the proportion of houses in the sample that are appraised for higher than the market prices is 0.24. -Referring to Instruction 8-15,what will be the 90% confidence interval for the total market price of the houses in the suburb constructed by the consulting group?

Other things being equal,as the confidence level for a confidence interval increases,the width of the interval increases.

Instruction 8-16 The Registrar of a university would like to estimate the proportion of the student population who owns a personal computer. In a sample of 500 students, 417 own a personal computer. -Referring to Instruction 8-16,the parameter of interest is the average number of students in the population who own a personal computer.

Instruction 8-19 A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students. The results were 1,344 answered "YES" and 1,456 answered "NO". -Referring to Instruction 8-19,the parameter of interest is the proportion of student population who feel comfortable reporting cheating by their fellow students.

Instruction 8-16 The Registrar of a university would like to estimate the proportion of the student population who owns a personal computer. In a sample of 500 students, 417 own a personal computer. -Referring to Instruction 8-16,the parameter of interest is the proportion of student population who own a personal computer.

Instruction 8-4 To become an actuary, it is necessary to pass a series of 10 exams, including the most important one, an exam in probability and statistics. An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program. They take a sample of 8 actuarial students in this program and determine that their scores are: 2, 5, 8, 8, 7, 6, 5, and 7. This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program. -Referring to Instruction 8-4,if we use the same sample information to obtain a 95% confidence interval,the resulting interval would be narrower than the one obtained here with 90% confidence.

Instruction 8-14 After an extensive advertising campaign, the manager of a company wants to estimate the proportion of potential customers that recognise a new product. She samples 120 potential consumers and finds that 54 recognise this product. She uses this sample information to obtain a 95% confidence interval that goes from 0.36 to 0.54. -Referring to Instruction 8-14,this interval requires the use of the t distribution to obtain the confidence coefficient.

Instruction 8-18 A poll was conducted by the marketing department of a video game company to determine the popularity of a new game that was targeted to be launched in three months. Telephone interviews with 1,500 young adults were conducted which revealed that 49% said they would purchase the new game. The margin of error was ±3 percentage points. -Referring to Instruction 8-18,the size of the population is 1,500.

Given a sample mean of 2.1 and a sample standard deviation of 0.7 from a sample of 10 data points,a 90% confidence interval will have a width of 2.36.

Instruction 8-7 A hotel chain wants to estimate the average number of rooms rented daily in each month. The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 24 rooms. -Referring to Instruction 8-7,during February,a sample of 25 days has a sample mean of 37 rooms.Use this information to calculate a 92% confidence interval for the population mean.

Instruction 8-18 A poll was conducted by the marketing department of a video game company to determine the popularity of a new game that was targeted to be launched in three months. Telephone interviews with 1,500 young adults were conducted which revealed that 49% said they would purchase the new game. The margin of error was ±3 percentage points. -Referring to Instruction 8-18,what is the needed sample size to obtain a 95% confidence interval estimate of the percentage of the targeted young adults who will purchase the new game by allowing the same level of margin of error?