9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

2:47 minutes

Problem 11.1.5

Textbook Question

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?

Verified step by step guidance

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.

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.

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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Textbook Question

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Textbook Question

Cybersecurity The table below lists the frequency of leading digits of Internet traffic interarrival times for a computer, along with the percentages of each leading digit expected with Benford’s law.

a. Identify the general notation used for observed and expected values.

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Textbook Question

b. Identify the observed and expected values for the leading digit of 2.

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Textbook Question

c. Use the results from part (b) to find the contribution to the x2 test statistic from the category representing the leading digit of 2.

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Textbook Question

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? Repeat the preceding exercise using the frequencies in the following table, which summarizes all of the 2784 male heights listed in Data Set 4 “Measured and Reported.” Does the larger data set have much of an effect on the results from Exercise 5?

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Textbook Question

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.

Testing a Slot Machine The author purchased a slot machine (Bally Model 809) and tested it by playing it 1197 times. There are 10 different categories of outcomes, including no win, win jackpot, win with three bells, and so on. When testing the claim that the observed outcomes agree with the expected frequencies, the author obtained a test statistic of x2 = 8.815 Use a 0.05 significance level to test the claim that the actual outcomes agree with the expected frequencies. Does the slot machine appear to be functioning as expected?

Textbook Question

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.

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Textbook Question

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.

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