FAD1015: Mathematics III — Tutorial 11

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

Topic: Hypothesis Testing About the Mean

Question 1

A manufacturer of detergent claims that the mean weight of a particular box of detergent is 3kg. A random sample of 100 boxes revealed a sample average of 2.75 kg and a sample standard deviation of 0.15kg.

(a) Using a 0.01 level of significance, is there evidence that the average weight of the boxes is different from 3kg?

(b) Construct a 95% confidence interval estimate of the population mean weight of the boxes.

Question 2

A manufacturer claims the life of their battery type has a mean of 54 months and a standard deviation of 6 months. A consumer group purchases a sample of 50 of these batteries and tests them. They find that the average life of the battery is 52 months. What should they conclude at $\alpha = 0.05$?

Question 3

A pain reliever currently being used in a hospital is known to bring relief to patients in a mean time of 3.5 minutes. To compare a new pain reliever with the one currently being used, the new drug is administered to a random sample of 50 patients. The mean time to relief for the sample of patients is 2.8 minutes and the standard deviation is 1.1 minutes.

(a) Do the data provide sufficient evidence to conclude that the new drug effectively reduces the mean time until a patient feels relief from pain?

(b) Construct a 95% confidence interval estimate of the population mean time to relief. Based on this computed confidence interval, test the hypothesis that the real mean time to relief is 3.5 minutes.

Question 4

28 smokers were questioned about the number of hours they sleep daily. We want to test the hypothesis that smokers need less sleep than the general public, which needs an average of 7.7 hours of sleep.

(a) Compute a rejection region for the significance level of 0.05.

(b) If the sample mean is 7.5, and the sample standard deviation is 0.5, what can you conclude?

Question 5

A company that sells frozen shrimp prints 'content: 12 ounces' on each package. It is undesirable to have the average contents be either less or greater than 12 ounces. A random sample of 25 packages yields an average of 11.83 ounces and a standard deviation of 0.5 ounces.

Using $\alpha = 0.05$, what conclusion should be drawn concerning the mean distribution of package weights?

Question 6

Tooth decay generally develops first on teeth with irregular shapes (typically molars). The most susceptible surfaces on these teeth are the chewing surfaces. Usually, the enamel on these surfaces contains tiny pockets that tend to hold food particles. Bacteria begin to eat the food particles to create an environment in which the tooth surface will decay. Of particular importance in the decay rate of teeth, in addition to the natural hardness of the teeth, is the form of food eaten by the individual. Some forms of carbohydrates are particularly detrimental to dental health. Many studies have been conducted to verify these findings, and we can imagine how a study might have been run. A random sample of 60 adults was obtained from a given locale. Each person was examined and then maintained a diet supplemented with a sugar solution at all meals. At the end of a 1-year period, the average number of newly decayed teeth for the group was 0.70, and the standard deviation was 0.4.

(a) Do these data present sufficient evidence ($\alpha = 0.05$) that the mean number of newly decayed teeth for the people whose diet includes a sugar solution is greater than 0.30? The value 0.30 was of interest as this is the rate that had been shown to apply to a person whose diet did not contain a sugar supplement.

(b) Why would a two-tailed test be inappropriate?

Question 7

A study was conducted on 90 adult male patients following a new treatment for congestive heart failure. One of the variables measured on the patients was the increase in exercise capacity (in minutes) over a 4-week treatment period. The previous treatment regime had produced an average increase of $\mu = 2$ minutes. The researchers wanted to evaluate whether the new treatment had increased the value of $\mu$ in comparison to the previous treatment. The data yielded $\bar{x} = 2.17$ and $s = 1.05$. Using $\alpha = 0.05$, what conclusions can you draw about the research hypothesis?

Question 8

Researchers are interested in the mean age of a certain population. A random sample of 10 individuals is drawn from the population of interest and has a mean of 27 years. Assuming that the population is approximately normally distributed with a variance of 20 years, can we conclude that the mean is different from 30 years at $\alpha = 0.05$? If the p-value is 0.0340, how can we use it in making a decision?

Question 9

A tobacco company advertises that the average nicotine content of its cigarettes is at most 14 milligrams. A consumer protection agency wants to determine whether the average nicotine content is in fact, greater than 14. A random sample of 300 cigarettes of the company's brand yielded an average nicotine content of 14.6 and a standard deviation of 3.8 milligrams. Determine the level of significance of the statistical test of the agency's claim that $\mu$ is greater than 14. If $\alpha = 0.01$, is there significant evidence that the agency's claim has been supported by the data?

Question 10

Among 157 African American men, the mean systolic blood pressure was 146 mmHg with a standard deviation of 27 mmHg. We wish to know if, based on these data, we may conclude that the mean systolic blood pressure for a population of African Americans is greater than 140 mmHg. At $\alpha = 0.01$ significance level, test using:

(a) Traditional method (critical value/rejection region)

(b) P-value method

(c) Confidence Interval method

Related Concepts

  • Hypothesis Testing — overview of statistical hypothesis testing framework
  • Null Hypothesis — statement of no effect or no difference (H₀)
  • Alternative Hypothesis — statement to be tested (H₁)
  • Significance Level — probability of Type I error (α)
  • P-Value — probability of observing test statistic at least as extreme
  • Type I Error — rejecting true null hypothesis
  • Type II Error — failing to reject false null hypothesis
  • Test Statistic — standardized value for hypothesis test
  • Critical Value — threshold for rejecting null hypothesis
  • Rejection Region — range of values leading to rejection of H₀
  • One-Tailed Test — directional hypothesis test
  • Two-Tailed Test — non-directional hypothesis test
  • Probability Distributions — normal and t-distributions for testing

Related Lectures

Related Course Page

Source: FAD1015 25-26 Tutorial 11 Questions.pdf