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Ch. 2 - Exploring Data with Tables and Graphs
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 2 - Exploring Data with Tables and GraphsProblem 2.1.28
Chapter 2, Problem 2.1.28

Births Natural births randomly selected from four hospitals in New York State occurred on the days of the week (in the order of Monday through Sunday) with these frequencies: 52, 66, 72, 57, 57, 43, 53. Does it appear that such births occur on the days of the week with equal frequency?

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that births occur with equal frequency across all days of the week, while the alternative hypothesis states that births do not occur with equal frequency.
Step 2: Calculate the expected frequency for each day of the week under the assumption of equal frequency. Since there are 7 days in a week and the total number of births is the sum of the given frequencies, divide the total number of births by 7 to find the expected frequency for each day.
Step 3: Use the chi-square test formula to calculate the test statistic. The formula is χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ represents the observed frequency for each day and Eᵢ represents the expected frequency for each day. Perform this calculation for all 7 days and sum the results.
Step 4: Determine the degrees of freedom for the chi-square test. The degrees of freedom are calculated as (number of categories - 1). In this case, there are 7 days, so the degrees of freedom are 7 - 1 = 6.
Step 5: Compare the calculated chi-square test statistic to the critical value from the chi-square distribution table at the chosen significance level (e.g., α = 0.05) and degrees of freedom. If the test statistic exceeds the critical value, 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.

Chi-Square Test of Goodness of Fit

The Chi-Square Test of Goodness of Fit is a statistical method used to determine if observed frequencies differ from expected frequencies. In this context, it helps assess whether the number of births across the days of the week is uniformly distributed or if certain days have significantly more or fewer births than expected.
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Step 2: Calculate Test Statistic

Null Hypothesis

The null hypothesis is a statement that assumes no effect or no difference, serving as a starting point for statistical testing. In this scenario, the null hypothesis would state that births occur with equal frequency across all days of the week, which can be tested against the observed data.
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Step 1: Write Hypotheses

Expected Frequencies

Expected frequencies are the theoretical frequencies that would occur if the null hypothesis were true. For this question, if births are equally likely on each day, the expected frequency for each day can be calculated by dividing the total number of births by the number of days (7), providing a benchmark for comparison with the observed frequencies.
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Creating Frequency Polygons
Related Practice
Textbook Question

In Exercises 5–8, identify the class width, class midpoints, and class boundaries for the given frequency distribution. Also identify the number of individuals included in the summary. The frequency distributions are based on real data from Appendix B.

7.

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Textbook Question

Freshman 15 Refer to Data Set 13 “Freshman 15” and use the second column, which lists weights (kg) in September of college freshmen. Begin with a lower class limit of 40 kg and use a class width of 10 kg. Does the distribution appear to be a normal distribution?

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Textbook Question

Analysis of Last Digits Weights of respondents were recorded as part of the California Health Interview Survey. The last digits of weights from 50 randomly selected respondents are listed below. Construct a frequency distribution with 10 classes. Based on the distribution, do the weights appear to be reported or actually measured? Does there appear to be a gap in the frequencies and, if so, how might that gap be explained? What do you know about the accuracy of the results?

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Textbook Question

In Exercises 9–18, construct the histograms and answer the given questions.

Chicago Commute Time Use the frequency distribution from Exercise 13 in Section 2-1 to construct a histogram. Does it appear to be the graph of data from a population with a normal distribution?

185
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Textbook Question

In Exercises 5–8, identify the class width, class midpoints, and class boundaries for the given frequency distribution. Also identify the number of individuals included in the summary. The frequency distributions are based on real data from Appendix B.

8.

570
views
Textbook Question

In Exercises 9–18, construct the histograms and answer the given questions.


Old Faithful Use the frequency distribution from Exercise 15 in Section 2-1 to construct a histogram. Does it appear to be the graph of data from a population with a normal distribution?

147
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