SECTION 10.1 Goodness-of-Fit Test 527 To calculate the test statistic for the chi-square goodness-of-fit test, you can use observed frequencies and expected frequencies. To calculate the expected frequencies, you must assume the null hypothesis is true. The observed frequency O of a category is the frequency for the category observed in the sample data. The expected frequency E of a category is the calculated frequency for the category. Expected frequencies are found by using the expected (or hypothesized) distribution and the sample size. The expected frequency for the ith category is Ei = npi where n is the number of trials (the sample size) and pi is the assumed probability of the ith category. DEFINITION Finding Observed Frequencies and Expected Frequencies A tax preparation company randomly selects 300 adults and asks them how they prepare their taxes. The results are shown at the right. Find the observed frequency and the expected frequency (using the distribution on the preceding page) for each tax preparation method. (Adapted from National Retail Federation) SOLUTION The observed frequency for each tax preparation method is the number of adults in the survey naming a particular tax preparation method. The expected frequency for each tax preparation method is the product of the number of adults in the survey and the assumed probability that an adult will name a particular tax preparation method. The observed frequencies and expected frequencies are shown in the table below. Tax preparation method % of people Observed frequency Expected frequency Accountant 24% 60 30010.242 = 72 By hand 20% 43 30010.202 = 60 Computer software 35% 117 30010.352 = 105 Friend or family 6% 29 30010.062 = 18 Tax preparation service 15% 51 30010.152 = 45 TRY IT YOURSELF 1 The tax preparation company in Example 1 decides it wants a larger sample size, so it randomly selects 500 adults. Find the expected frequency for each tax preparation method for n = 500. Answer: Page A42 The sum of the expected frequencies always equals the sum of the observed frequencies. For instance, in Example 1 the sum of the observed frequencies and the sum of the expected frequencies are both 300. EXAMPLE 1 Survey results 1n = 3002 Accountant 60 By hand 43 Computer software 117 Friend or family 29 Tax preparation service 51 Picturing the World The pie chart shows the distribution of health care visits to doctor offices, emergency departments, and home visits in a recent year. (Source: National Center for Health Statistics) 1–3 visits 49.2% None 14.5% 4–9 visits 23.4% 10 or more visits 12.9% A researcher randomly selects 200 people and asks them how many visits they make to the doctor in a year: 1–3, 4–9, 10 or more, or none. What is the expected frequency for each response?
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