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Ch. 11 - Goodness-of-Fit and Contingency Tables
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 11 - Goodness-of-Fit and Contingency TablesProblem 11.1.5
Chapter 11, Problem 11.1.5

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.


Heights Measured or Reported? A random sample of the last digits of heights (in.) of males from Data Set 4 “Measured and Reported” is summarized in the table below. Use these last digits to determine whether they occur with about the same frequency. Use a 0.05 significance level. Do the corresponding heights appear to be measured or reported?


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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀: The last digits occur with about the same frequency (uniform distribution). H₁: The last digits do not occur with the same frequency (not uniform).
Step 2: Calculate the expected frequency for each digit under the assumption of uniform distribution. Since there are 10 digits (0 through 9) and the total frequency is the sum of all observed frequencies, divide the total frequency by 10 to get the expected frequency for each digit.
Step 3: Use the Chi-Square test formula to calculate the test statistic. The formula is: χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ is the observed frequency and Eᵢ is the expected frequency for each digit.
Step 4: Determine the degrees of freedom (df) for the Chi-Square test. The formula for degrees of freedom is: df = k - 1, where k is the number of categories (digits in this case).
Step 5: Compare the calculated test statistic to the critical value from the Chi-Square distribution table at the 0.05 significance level and the appropriate degrees of freedom. Alternatively, calculate the P-value and compare it to the significance level. Based on this comparison, decide whether to reject or fail to reject the null hypothesis and state the conclusion about whether the heights appear to be measured or reported.

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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 no difference, and an alternative hypothesis (H1) that indicates the presence of an effect. The test assesses the evidence against H0 using a test statistic and a significance level, leading to a conclusion about whether to reject or fail to reject H0.
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Step 1: Write Hypotheses

Chi-Square Test

The Chi-Square test is a statistical test used to determine if there is a significant association between categorical variables. In this context, it can be applied to assess whether the observed frequencies of last digits of heights differ from expected frequencies, which would indicate whether the heights are measured or reported. The test calculates a Chi-Square statistic, which is then compared to a critical value from the Chi-Square distribution to draw conclusions.
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Step 2: Calculate Test Statistic

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis in hypothesis testing. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming that the null hypothesis is true. A smaller P-value indicates stronger evidence against H0, and if it is less than the significance level (e.g., 0.05), the null hypothesis is rejected.
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Step 3: Get P-Value
Related Practice
Textbook Question

Right-Tailed, Left-Tailed, Two-Tailed Is the hypothesis test described in Exercise 1 right-tailed, left-tailed, or two-tailed? Explain your choice.

Textbook Question

Accuracy of Fingerprint Identifications An experiment was conducted to compare the accuracy of fingerprint experts to the accuracy of novices (based on data from “Identifying Fingerprint Expertise,” by Tangen, Thompson, and McCarthy, Psychological Science, Vol. 22, No. 8). The data in the table are based on trials in which the evaluators were given matching fingerprints. Use a 0.05 significance level to determine whether correct identification is independent of whether the evaluator is an expert or a novice.


Textbook Question

Gender and Eye Color The following table describes the distribution of eye colors reported by male and female statistics students (based on data from “Does Eye Color Depend on Gender? It Might Depend on Who or How You Ask,” by Froelich and Stephenson, Journal of Statistics Education, Vol. 21, No. 2). Is there sufficient evidence to warrant rejection of the belief that gender and eye color are independent traits? Use a 0.01 significance level.


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Textbook Question

Ghosts The following table summarizes results from a Pew Research Center survey in which subjects were asked whether they had seen or been in the presence of a ghost. Use a 0.01 significance level to test the claim that gender is independent of response. Does the conclusion change if the significance level is changed to 0.05?