FAD1015: Mathematics III — Tutorial 10

Centre for Foundation Studies in Science
Universiti Malaya
Session 2024/2025

Topic: Estimation of the Population Mean

Question 1

If $\bar{X} = 125$, $\sigma = 24$ and n=36, construct a 99% confidence interval estimate for the population mean, $\mu$. Interpret the interval.

Question 2

Is it possible to have a 100% confidence interval of the population mean? Why?

Question 3

A food-testing laboratory in Abu Dhabi has been commissioned to carry out an estimate of the amount of olive oil contained in 1-gallon tins purchased from a range of international olive oil producers. The manufacturers state that the standard deviation of the amount of olive oil is equal to 0.02 gallons. A random sample of 50 tins is selected, and the mean sample amount of olive oil per 1-gallon tin is 0.995 gallons.

(a) Construct a 99% confidence interval estimate of the population mean amount of olive oil and interpret its meaning.

(b) Based on these results, do you think retailers and consumers have a right to complain to the manufacturers? Why?

(d) Construct a 95% confidence interval estimate. How does this change your answer to b?

Question 4

A cereal company randomly selects 25 boxes (12 oz per box) of Cheerios every 10 minutes and weighs the boxes. Suppose the weights have a normal distribution with $\sigma = 0.2$ oz. One such sample yields $\bar{x} = 12.3$ oz.

(a) Calculate a 95% confidence interval for the mean weight $\mu$ of the packages produced during the period of time from which the sample was selected.

(b) Give a careful nonstatistical jargon interpretation of the confidence interval.

(c) The process engineer at the cereal company is concerned that the confidence intervals for $\mu$ are too wide to be of practical use.

i) If we double the sample size from 25 to 50, what is the impact on the width of the 95% confidence intervals?

ii) If we increase the level of confidence from 95% to 99%, what is the impact on the width of the confidence intervals?

(d) Because the company is collecting samples containing 25 boxes every 10 minutes, there are 720 confidence intervals constructed during every 5-day period.

i) If the level of confidence is 95% for each of the 720 intervals, how many of the intervals would you expect to be in error—that is, fail to contain $\mu$?

ii) If the sample size is kept at 25 boxes per sample, but the level of confidence is increased from 95% to 99%, how many of the 95% confidence intervals would you expect to be in error during any 5-day period?

Question 5

The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The standard deviation is 100 hours. A random sample of 64 light bulbs indicated a sample mean life of 350 hours.

(a) Construct a 95% confidence interval estimate for the population mean life of light bulbs in this shipment.

(b) Do you think the manufacturer has the right to state that the light bulbs have a mean life of 400 hours? Why?

(d) Suppose that the standard deviation changes to 80 hours. What are your answers in (a) and (b)?

Question 6

T&J courier company in PJ claims that its mean delivery time to any place in the city is less than 3 hours. To evaluate the claim, the quality control personnel randomly select 50 deliveries and compute the mean delivery time to be $\bar{x} = 2.8$ hours with a standard deviation $s = 0.6$ hours.

(a) Estimate the mean delivery time $\mu$ using a 95% confidence interval.

(b) Based on the 95% confidence interval, does the company's claim appear reasonable?

Question 7

The research department in a cosmetic company needs to estimate the average effect duration of a new skin cream. The population standard deviation is unknown. A random sample of 25 test subjects reported an average effect duration of 8 hours.

(a) Construct a 95% confidence interval estimate for the average effect duration of the skin cream. Assume the sample standard deviation is 1.5 hours.

(b) Based on your confidence interval, is it justified for the company to claim that the cream's effect lasts on average 9 hours? Discuss your reasoning.

Question 8

Assuming that the population is normally distributed, construct a 95% confidence interval estimate for the population mean for each of the following samples:

Sample A: 1 1 1 1 8 8 8 8

Sample B: 1 2 3 4 5 6 7 8

Explain why these 2 samples produce different confidence intervals despite having the same mean and range.

Question 9

A stationery store wants to estimate the mean retail value of greeting cards that it has in its inventory. A random sample of 100 greeting cards indicates a mean value of $2.55 and a standard deviation of $0.44

(a) Assuming a normal distribution, construct a 95% confidence interval estimate for the mean value of all greeting cards in the store's inventory.

(b) Suppose there were 2,500 greeting cards in the store's inventory. How are the results in (a) helpful in assisting the store owner in estimating the total value of the inventory?

Question 10

The Chamber of Commerce in a city wants to estimate the gross profit margin of small businesses (under $500,000 in sales) in their city. A random sample of the year-end statements of 10 small businesses shows the mean gross profit margin to be 5.2% (of sales) with a standard deviation of 7.5%.

(a) Construct a 99% confidence interval for the mean gross profit margin $\mu$ of all small businesses in the city.

(b) What are some limitations in using the confidence interval you constructed in (a)? For example, since the sample size is small, do you think that the data come from a normal distribution? Is it valid to replace $\sigma$ with $s$?

Related Concepts

Confidence Interval — range of values likely to contain population parameter
Point Estimate — single value estimate of population parameter
T-Distribution — used when population standard deviation is unknown
Margin of Error — half-width of confidence interval
Standard Error — standard deviation of the sampling distribution
Degrees of Freedom — parameter for t-distribution
Level of Confidence — probability that interval contains true parameter

Source: FAD1015 25-26 Tutorial 10 Questions.pdf