Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 9 - Inferences from Two Samples

Problem 9.1.15c

Chapter 9, Problem 9.1.15c

Can Dogs Detect Malaria? A study was conducted to determine whether dogs could detect malaria from socks worn by malaria patients and socks worn by patients without malaria. Among 175 socks worn by malaria patients, the dogs made correct identifications 123 times. Among 145 socks worn by patients without malaria, the dogs made correct identifications 131 times (based on data presented at an annual meeting of the American Society of Tropical Medicine, by principal investigator Steve Lindsay). Use a 0.05 significance level to test the claim of no difference between the two rates of correct responses.

c. What do the results suggest about the use of dogs to detect malaria?

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that there is no difference between the rates of correct identifications for socks worn by malaria patients and socks worn by patients without malaria. Mathematically, H₀: p₁ = p₂. The alternative hypothesis states that there is a difference between the rates, H₁: p₁ ≠ p₂.

Step 2: Calculate the sample proportions for each group. For socks worn by malaria patients, the proportion is p₁ = 123 / 175. For socks worn by patients without malaria, the proportion is p₂ = 131 / 145.

Step 3: Compute the pooled proportion (p̂) since we are testing the claim of no difference. The pooled proportion is calculated as p̂ = (x₁ + x₂) / (n₁ + n₂), where x₁ and x₂ are the number of correct identifications for each group, and n₁ and n₂ are the total number of socks in each group.

Step 4: Calculate the test statistic using the formula for the z-test for two proportions: z = (p₁ - p₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]. Substitute the values of p₁, p₂, p̂, n₁, and n₂ into the formula to compute the z-value.

Step 5: Compare the calculated z-value to the critical z-value at the 0.05 significance level for a two-tailed test. If the absolute value of the z-value exceeds the critical z-value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the results in the context of the problem to determine whether the data suggest that dogs can detect malaria effectively.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) that represents no effect or difference, and an alternative hypothesis (H1) that indicates the presence of an effect or difference. In this case, the null hypothesis would state that there is no difference in the detection rates of malaria between the two groups of socks.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. A common significance level is 0.05, which implies that there is a 5% risk of concluding that a difference exists when there is none. In this study, using a 0.05 significance level means that if the p-value obtained from the test is less than 0.05, the results would be considered statistically significant.

P-Value

The p-value is a statistical measure that helps determine the strength of the evidence against the null hypothesis. It quantifies the probability of observing the sample data, or something more extreme, if the null hypothesis is true. A low p-value (typically less than the significance level) indicates strong evidence against the null hypothesis, suggesting that the dogs' detection abilities may differ significantly between the two groups of socks.

Recommended video:

Guided course

06:50

Step 3: Get P-Value

Related Practice

Textbook Question

Overlap of Confidence Intervals In the article “On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals,” by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement: “Independent simple random samples, each of size 200, have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute.”

c. Use a 0.05 significance level to test the claim that the two population proportions are equal. What do you conclude?

Textbook Question

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1-1 and n2-1)

Magnet Treatment of Pain People spend around \$5 billion annually for the purchase of magnets used to treat a wide variety of pains. Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results given below are among the results obtained in the study (based on data from “Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study,” by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10). Higher scores correspond to greater pain levels.

c. Does it appear that magnets are effective in treating back pain? Is it valid to argue that magnets might appear to be effective if the sample sizes are larger?

" style="max-width: 100%; white-space-collapse: preserve;" width="550">

Textbook Question

In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.

Heights of Presidents A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (cm) of presidents along with the heights of their main opponents (from Data Set 22 “Presidents” in Appendix B).

b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

Textbook Question

Confidence Interval Assume that we want to use the sample data in Exercise 1 for constructing a confidence interval to be used for testing the given claim.

c. If the resulting confidence interval is -5.8 admissions <ud < -0.9 admissions, what do you conclude?

Textbook Question

F Test Statistic

c. If testing the claim that sigma2,1 is not equals to sigma2,2 what do we know about the two samples if the test statistic F is very close to 1?

Textbook Question

Can Dogs Detect Malaria? A study was conducted to determine whether dogs could detect malaria from socks worn by malaria patients and socks worn by patients without malaria. Among 175 socks worn by malaria patients, the dogs made correct identifications 123 times. Among 145 socks worn by patients without malaria, the dogs made correct identifications 131 times (based on data presented at an annual meeting of the American Society of Tropical Medicine, by principal investigator Steve Lindsay). Use a 0.05 significance level to test the claim of no difference between the two rates of correct responses.

b. Test the claim by constructing an appropriate confidence interval.