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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 9 - Inferences from Two SamplesProblem 9.1.15b
Chapter 9, Problem 9.1.15b

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.

Verified step by step guidance
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Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that there is no difference between the two rates of correct responses (p₁ = p₂). The alternative hypothesis (H₁) states that there is a difference between the two rates of correct responses (p₁ ≠ p₂).
Step 2: Calculate the sample proportions for each group. For the malaria patients, the sample proportion is p̂₁ = x₁ / n₁, where x₁ = 123 and n₁ = 175. For the non-malaria patients, the sample proportion is p̂₂ = x₂ / n₂, where x₂ = 131 and n₂ = 145.
Step 3: Compute the pooled proportion under the null hypothesis. The pooled proportion is calculated as p̂ = (x₁ + x₂) / (n₁ + n₂), where x₁ and x₂ are the number of correct identifications, and n₁ and n₂ are the total sample sizes for each group.
Step 4: Calculate the test statistic for the difference in proportions. The test statistic is given by z = (p̂₁ - p̂₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)], where p̂₁ and p̂₂ are the sample proportions, p̂ is the pooled proportion, and n₁ and n₂ are the sample sizes.
Step 5: Construct a confidence interval for the difference in proportions. The confidence interval is given by (p̂₁ - p̂₂) ± z* × √[p̂₁(1 - p̂₁)/n₁ + p̂₂(1 - p̂₂)/n₂], where z* is the critical value for the desired confidence level (e.g., 1.96 for 95%). Compare the confidence interval to determine if it includes 0, which would support the null hypothesis.

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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. The goal is to determine whether the observed data provide sufficient evidence to reject the null hypothesis at a specified significance level, such as 0.05.
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Step 1: Write Hypotheses

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95% or 99%. It provides an estimate of uncertainty around a sample statistic, such as a mean or proportion. In the context of the question, constructing a confidence interval for the difference in detection rates will help assess whether there is a statistically significant difference between the two groups of socks.
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Introduction to Confidence Intervals

Proportions and Sample Size

Proportions represent the fraction of a whole, often used in statistics to describe the success rate of an event occurring within a sample. In this study, the proportions of correct identifications by dogs for both malaria and non-malaria socks are crucial for analysis. Additionally, understanding sample size is important, as larger samples generally provide more reliable estimates and greater statistical power to detect differences between groups.
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Sampling Distribution of Sample Proportion
Related Practice
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.


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

Textbook Question

Second-Hand Smoke Samples from Data Set 15 “Passive and Active Smoke” include cotinine levels measured in a group of smokers ( n = 40, x_bar = 172.48 ng/mL, 119.50 ng/mL ) and a group of nonsmokers not exposed to tobacco smoke ( n = 40, x_bar = 16.35 ng/mL, 62.53 ng/mL ). Cotinine is a metabolite of nicotine, meaning that when nicotine is absorbed by the body, cotinine is produced.


b. The 40 cotinine measurements from the nonsmoking group consist of these values (all in ng/mL): 1, 1, 90, 244, 309, and 35 other values that are all 0. Does this sample appear to be from a normally distributed population? If not, how are the results from part (a) affected?

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.


The Freshman 15 The “Freshman 15” refers to the belief that college students gain 15 lb (or 6.8 kg) during their freshman year. Listed below are weights (kg) of randomly selected male college freshmen (from Data Set 13 “Freshman 15” in Appendix B). The weights were measured in September and later in April.


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

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

Independent Samples Which of the following involve independent samples?


b. Data Set 6 “Births” includes birth weights of a sample of baby boys and a sample of baby girls.


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?