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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 9.2.27
Chapter 9, Problem 9.2.27

No Variation in a Sample An experiment was conducted to test the effects of alcohol. Researchers measured the breath alcohol levels for a treatment group of people who drank ethanol and another group given a placebo. The results are given below (based on data from “Effects of Alcohol Intoxication on Risk Taking, Strategy, and Error Rate in Visuomotor Performance,” by Streufert et al., Journal of Applied Psychology, Vol. 77, No. 4). Use a 0.05 significance level to test the claim that the two sample groups come from populations with the same mean.


"Data: Treatment group n=22, mean=0.049, sd=0.015; Placebo group n=22, mean=0.000, sd=0.000."

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Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that the two sample groups come from populations with the same mean (μ₁ = μ₂). The alternative hypothesis (H₁) states that the two sample groups come from populations with different means (μ₁ ≠ μ₂).
Step 2: Determine the appropriate statistical test. Since we are comparing the means of two independent groups, and the standard deviation for the placebo group is zero (s₂ = 0), this suggests no variation in the placebo group. A t-test for independent samples may not be appropriate due to the lack of variability in one group. Instead, consider using a non-parametric test like the Mann-Whitney U test or consult the assumptions of the t-test.
Step 3: Calculate the test statistic. If using a t-test, the formula for the test statistic is: t = (x̄₁ - x̄₂) / sqrt((s₁²/n₁) + (s₂²/n₂)). Substitute the given values: x̄₁ = 0.049, x̄₂ = 0.000, s₁ = 0.015, s₂ = 0.000, n₁ = 22, n₂ = 22. Note that s₂ = 0 will affect the calculation.
Step 4: Determine the degrees of freedom. For a t-test, the degrees of freedom can be calculated using the formula: df = n₁ + n₂ - 2. Substitute the values: n₁ = 22, n₂ = 22.
Step 5: Compare the test statistic to the critical value or p-value at the 0.05 significance level. If the test statistic exceeds the critical value or the p-value is less than 0.05, reject the null hypothesis. Otherwise, fail to reject 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 population parameters based on sample data. It involves formulating a null hypothesis (H0) that assumes no effect or difference, and an alternative hypothesis (H1) that suggests a significant effect or difference. In this case, the null hypothesis would state that the mean breath alcohol levels of the treatment and placebo groups are equal.
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Step 1: Write Hypotheses

Significance Level

The significance level, denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. A common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In this experiment, using a 0.05 significance level means that if the p-value obtained from the test is less than 0.05, the null hypothesis can be rejected, suggesting a significant difference between the groups.
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Step 4: State Conclusion Example 4

T-test for Independent Samples

A t-test for independent samples is used to compare the means of two groups to determine if they are statistically different from each other. This test takes into account the sample sizes, means, and standard deviations of both groups. Given the data provided, the t-test will help assess whether the observed difference in breath alcohol levels between the treatment and placebo groups is significant, based on the calculated t-statistic and corresponding p-value.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

Body Temperatures Listed below are body temperatures from six different subjects measured at two different times in a day (from Data Set 5 “Body Temperatures” in Appendix B).


a. Are the two sets of data independent or dependent? Explain.


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Textbook Question

Test for Normality For the hypothesis test described in Exercise 2, the sample sizes are n1 = 2208 and n2 = 1986 When using the F test with these data, is it correct to reason that there is no need to check for normality because both samples have sizes that are greater than 30?

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Textbook Question

Randomization vs t Test Two samples of commute times from Boston and New York are randomly selected and it is found that the samples sizes are n1 = 18 and n2 = 12 and each of the two samples appears to be from a population with a distribution that is dramatically far from normal. Which method is more likely to yield better results for testing Mu1 is not equals to Mu2. Hypothesis test using the t distribution (as in Section 9-2) or the resampling method?

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Textbook Question

Robust What does it mean when we say that the F test described in this section is not robust against departures from normality?

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Textbook Question

Is Friday the 13th Unlucky? Listed below are numbers of hospital admissions in one region due to traffic accidents on different Fridays falling on the 6th day of a month and the following 13th day of the month (based on data from “Is Friday the 13th Bad for Your Health,” by Scanlon et al., British Medical Journal, Vol. 307). Assume that we want to use a 0.05 significance level to test the claim that the data support the claim that fewer hospital admissions due to traffic accidents occur on Friday the 6th than on the following Friday the 13th. Identify the null hypothesis and alternative hypothesis.


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Textbook Question

Degrees of Freedom In Exercise 20 “Blanking Out on Tests,” using the “smaller of n1-1 and n2-1” for the number of degrees of freedom results in df=15 Find the number of degrees of freedom using Formula 9-1. In general, how are hypothesis tests and confidence intervals affected by using Formula 9-1 instead of the “smaller of n1-1 and n2-1 ”?

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