Confidence Intervals
ODD numbers only
Show calculations
No Ai, Chegg, etc.
Example 1
A software development company wanted to estimate the population mean of monthly expenditures for their 250 software developers. Forty developers were randomly selected. Their mean monthly expenditures were RM5,000 with a population standard deviation of RM500.
a. What is the population mean?
b. Construct a 95% confidence interval for the population mean and interpret the result.
c. Construct a 99% confidence interval for the population mean and interpret the result.
d. Is it reasonable to conclude that the population mean will be RM5,200 at the 99% confidence level?
e. Compare the results of (b) and (c).
f. Is there any benefit of using a higher confidence level?
Example 2
The CGPAs for 10 randomly selected students are assumed to be normally distributed
2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8
Find a 98 percent confidence interval for the true mean. Can you conclude that the population mean is equal to 4? Explain.
Example 3
The American Sugar Producers Association wants to estimate the mean yearly sugar consumption. A sample of 16 people reveals the mean yearly consumption to be 60 pounds with a S.D of 20 pounds.
a. What is the value of the population mean? What is the best estimate of this value?
b. Why do we need to use the -distribution? What assumptions do you need to make?
c. For a 90% confidence interval, what is the value of ?
d. Develop the 90% confidence interval for the population mean.
e. Would it be reasonable to conclude that the population mean is 63 pounds?
Example 4
Of 369 randomly selected medical students, 23 said that they planned to work in a rural community. Construct a 95% confidence interval for the percentage of all medical students who plan to work in a rural community. Interpret your findings.
Example 5
In a survey carried out at IIUM Educare, 28 children out of a random sample 80 said that they read Kuntum magazine regularly.
a. Calculate the 95% confidence interval for the true proportion of all children who read this magazine at IIUM Educare.
b. A similar survey at IIUM Educare found that 45 children out of a random sample of 100 said they read Upin & Ipin comics regularly. Find the 95% confidence interval for the true proportion of all children who read this comic at IIUM Educare.
c. On the basis of these surveys, is there any evidence that the sales of Upin & Ipin comics are higher than the sales of Kuntum magazine at IIUM Educare? Justify your answer.
Example 6
There are 250 computer programmers employed at Computers.com, Inc. A sample of 50 programmers revealed that 30 graduated with a university degree. Construct the 98 percent confidence interval for the proportion of all programmers who graduated from a university.
Example 7
An auditor wants to examine the monetary error made by the purchasing order department for a car dealership in the month of July. He decided to randomly sample 100 of the 925 purchase orders for the month of July and found the amount of error in each one. The sample mean and sample standard deviation are $6.00 and $17.012, respectively. Estimate with 95% confidence the average amount of error per purchase order for the entire month of July.
Example 8
A processor of carrots cuts the green top off each carrot, washes the carrots and inserts six to a package. Twenty packages are inserted in a box for shipment. To test the weight of the boxes, a few were checked. The mean weight was 20.4 pounds, the standard deviation 0.5 pounds. How many boxes must the processor sample to be 95 percent confident that the sample mean does not differ from the population mean by more than 0.2 pounds?
Example 9
Past surveys reveal that 30 percent of tourists going to LA to gamble during a weekend spend more than $1000. Management wants to update this percentage.
a. The new study is to use the 90 percent confidence level. The estimate is to be within 1 percent of the population proportion. What is the necessary sample size?
b. Management said that the sample size determined is too large. What can be done to reduce the sample size?
c. Based on your suggestion above, recalculate the sample size.
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