STAT20029 (T2,2020) Questions: Week 8

Confidence Interval Estimation (Refer Chapter 8)

Activity 8.1

Activity 8.1: Find the missing values?

Activity 8.2

Activity 8.2: (a) Find the critical Z values for the 99% confidence level of confidence interval?

(b) Find the critical Z values for the 90% confidence level of confidence interval?

Practice Problem 8.2

PP8.2: A random sample of size 70 is taken from a population that has a variance of 49. The sample mean is 90.4. What is the point estimate of µ? Construct a 94% confidence interval for µ.

Practice Problem 8.4

PP8.4: A company sells 25 gram boxes of sultanas that are promoted as a healthy snack food option for children. The company wants to estimate the number of sultanas that is packed into a box. To do so, a random sample of 30 boxes of sultanas is selected during a production run. The number of sultanas in each box is counted. Using this sample, the average number of sultanas per box is calculated to be 50.3. Taking the standard deviation as being known to be 1.2 sultanas per box, what is the point estimate of the number of sultanas per box? Construct a 95% confidence interval to estimate the mean number of sultanas packed per box during the production process.

Practice Problem 8.5

PP8.5: The average total dollar purchase at a convenience store is less than that at a supermarket. Despite smaller purchases, convenience stores can still be profitable because of the size of operation, volume of business and the mark-up. A researcher is interested in estimating the average purchase amount for convenience stores in suburban Melbourne. To do so, she randomly samples 24 purchases from several convenience stores in suburban Melbourne and tabulates the amounts to the nearest dollar. Use the following data to construct a 90% confidence interval for the population average amount of purchases. Assume that the population standard deviation is 3.23 dollars and the population is normally distributed.

Activity 8.3

Activity 8.3: Find the missing DF and t – critical values for the given confidence interval?

Practice Problem 8.12

PP8.12: A random sample of 15 items is taken, producing a sample mean of 2.364 with a sample variance of .81. Assuming that x is normally distributed, construct a 90% confidence interval for the population mean.

Practice Problem 8.16

PP8.16: The marketing director of a large department store wants to estimate the average number of customers who enter the store every 5 minutes. She randomly selects 5-minute intervals and counts the number of arrivals at the store. She obtains the figures 58, 32, 41, 47, 56, 80, 45, 29, 32 and 78. The analyst assumes that the number of arrivals is normally distributed. Using these data, the analyst computes a 95% confidence interval to estimate the mean value for all 5-minute intervals. What interval values does she get?

Activity 8.4

Activity 8.4: A clothing company manufactures jeans of waist sizes 82, 87, 92 and 97 centimetres. Since the jeans is manufactured and made by machines, there is a slight margin of error in the waist sizes. If the standard deviation is 1.10 cm, estimate the 95% confidence interval for a lot of 12 jeans that has a label of 87 waist size. What assumption, if any, did you have to make?

Practice Problem 8.18(C)

8.18: Use the following sample information to calculate the confidence interval to estimate the population proportion. Let x be the number of items in the sample with the characteristic of interest. (c) n = 240 and x = 106, with 85% confidence

Practice Problem 8.21

PP8.21: VicRoads wants to estimate the proportion of vehicles on the Hume Highway between the hours of midnight and 5.00 am that are semitrailers. The estimate will be used to determine highway repair and construction considerations and in highway patrol planning. Suppose researchers for VicRoads counted vehicles at different locations on the highway for several nights during this time period. Of the 3481 vehicles counted, 927 were semitrailers.

(a) Determine the point estimate for the proportion of vehicles travelling the Hume Highway during this time period that are semitrailers.

(b) Construct a 99% confidence interval for the proportion of vehicles on the Hume Highway during this time period that are semitrailers.

Activity 8.5

Activity 8.5: In a survey of drug use among 960 Sydney teenagers, the following results were reported. Estimate the 90% confidence interval of the proportion of all Sydney teenagers who are daily smokers or occasional smokers.

Practice Problem 8.31

PP8.31: A bank manager wants to determine the average total monthly deposits per customer at the bank. He believes an estimate of this average amount using a confidence interval is sufficient. How large a sample should he take to be within $200 of the actual average with 99% confidence? He assumes that the standard deviation of total monthly deposits for all customers is about $1000.

Practice Problem 8.33

PP8.33: A group of investors wants to develop a chain of fast-food restaurants. In determining potential costs for each facility, they must consider, among other expenses, the average monthly electricity bill. They decide to sample some fast-food restaurants currently operating to estimate the monthly cost of electricity. They want to be 90% confident of their results and want the error of the interval estimate to be no more than $100. They estimate that such bills range from $600 to $2500. How large a sample should they take?

Practice Problem 8.34

PP8.34: Suppose a production facility purchases a particular component in large lots from a supplier. The production manager wants to estimate the proportion of defective parts received from this supplier. She believes the proportion defective is no more than .20 and wants to be within .02 of the true proportion of defective parts with a 90% level of confidence. How large a sample should she take?

Review Problem 8.14

RP8.14: A research company has been asked to determine the proportion of all restaurants in Western Australia that serve alcoholic beverages. The company wants to be 98% confident of its results but has no idea what the actual proportion is. The company would like to report an error of no more than .05. How large a sample should it take?

Activity 8.6

Activity 8.6: The Head of Data Science department is interested in estimating the proportion of students entering the department who will choose Internet of Things (IoT) as their major. Suppose there is no information about the proportion of students who might choose the option. What conservative sample size should the department head take if she wants to be 80% confident that the estimate is within 0.10 of the true proportion?