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Chapter 12: Problem 11

An article about the California lottery that appeared in the San Luis Obispo Tribune (December 15, 1999) gave the following information on the age distribution of adults in California: \(35 \%\) are between 18 and 34 years old, \(51 \%\) are between 35 and 64 years old, and \(14 \%\) are 65 years old or older. The article also gave information on the age distribution of those who purchase lottery tickets. The following table is consistent with the values given in the article: $$ \begin{array}{lc} \text { Age of Purchaser } & \text { Frequency } \\ \hline 18-34 & 36 \\ 35-64 & 130 \\ 65 \text { and over } & 34 \\ \hline \end{array} $$ Suppose that the data resulted from a random sample of 200 lottery ticket purchasers. Based on these sample data, is it reasonable to conclude that one or more of these three age groups buys a disproportionate share of lottery tickets? Use a chi-square goodness-of-fit test with \(\alpha=.05\).

Short Answer

Expert verified
Yes, it is reasonable to conclude that one or more of the age groups buys a disproportionate share of lottery tickets. This conclusion is drawn from the chi-square goodness-of-fit test, which yielded a chi-square value (25.6218) that is greater than the critical chi-square value (5.99).

Step by step solution

01

Calculate the Expected Frequencies

First, we need to compute the expected frequencies for each age group. We do this by multiplying the total sample size (200) by the proportion of each age group in the population. The expected frequencies are therefore: \[ E_{18-34} = 200 * 0.35 = 70 \] \[ E_{35-64} = 200 * 0.51 = 102 \] \[ E_{65+} = 200 * 0.14 = 28 \]
02

Calculate the Observed Frequencies

The observed frequencies are given in the problem: \[ O_{18-34} = 36 \] \[ O_{35-64} = 130 \] \[ O_{65+} = 34 \]
03

Calculate the Chi-Square Statistic

Next, we calculate the chi-square test statistic using the formula: \[ \chi^2 = \Sigma \frac{(O-E)^2}{E} \]This yields: \[ \chi^2 = \frac{(36-70)^2}{70} + \frac{(130-102)^2}{102} + \frac{(34-28)^2}{28} \] \[ \chi^2 = 16.5714 + 7.7647 + 1.2857 \] \[ \chi^2 = 25.6218 \]
04

Calculate the Degree of Freedom and Critical Chi-Square Value

The degree of freedom is calculated as k - 1, where k is the number of categories in the distribution. In this case, k = 3. Hence, degree of freedom(df) = 3 - 1 = 2. The critical value of chi-square for a degree of freedom = 2 and significance level 伪 = 0.05 is 5.99.
05

Interpret the Results

Since the calculated chi-square value (25.6218) is greater than the critical chi-square value (5.99), we reject the null hypothesis that there is no difference between the observed and expected frequencies. This indicates that it is likely that one or more of the age groups buys a disproportionate share of lottery tickets as compared to what would be expected based on their proportion in the general population.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Frequency
Expected frequency is a key concept in the chi-square goodness-of-fit test. It represents the number of observations we would expect to see in an ideal scenario where the sample matches the expected proportions of each group. To find the expected frequency for each group, we use the formula: \( E_i = N \times p_i \), where \( E_i \) is the expected frequency, \( N \) is the total number of observations in the sample, and \( p_i \) is the proportion of the group in the population.

For instance, in the provided exercise, the expected frequency for the age group 18-34 is calculated as \( 200 \times 0.35 = 70 \). This means that, based on the population distribution, we would expect 70 individuals in this age group to purchase lottery tickets out of a sample of 200. Similarly, calculations for the other age groups give us 102 expected purchasers for ages 35-64, and 28 for ages 65 and over. These figures are crucial as they provide a benchmark against which actual observations are compared.
Observed Frequency
Observed frequency refers to the actual count of occurrences within each category in the sample data. This is the data we collect or are given as part of our experiment or survey. Observed frequencies are compared against expected frequencies to determine if there is a significant discrepancy between what we expect and what we observe.

In the exercise about the California lottery, the observed frequencies provided are: 36 for the age group 18-34, 130 for the age group 35-64, and 34 for age group 65 and over. These observed values are compared against the expected frequencies to examine whether these ages purchase lottery tickets in proportion to their representation in the general population.

The observed frequency is a fundamental part of the chi-square calculation as it allows us to see if any observed values deviate significantly from the expected under the null hypothesis of no difference.
Degree of Freedom
The degree of freedom (df) is an important concept when conducting hypothesis tests like the chi-square goodness-of-fit test. It refers to the number of independent values or quantities that can be assigned to a statistical distribution after certain constraints or conditions are applied.

In the context of the chi-square test, the degree of freedom is calculated as one less than the number of categories observed, i.e., \( df = k - 1 \), where \( k \) is the number of distinct categories. For the lottery example, there are three age categories, so the degree of freedom is \( 3 - 1 = 2 \).

Knowing the degree of freedom helps us determine the critical value from the chi-square distribution table, which is needed to interpret the significance of our calculated chi-square statistic. In this exercise, a critical value of 5.99 with 2 degrees of freedom and a 0.05 significance level is used to assess whether the observed distribution significantly deviates from the expected, leading to a conclusion about purchasing patterns among different age groups.

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Most popular questions from this chapter

According to Census Bureau data, in 1998 the California population consisted of \(50.7 \%\) whites, \(6.6 \%\) blacks, \(30.6 \%\) Hispanics, \(10.8 \%\) Asians, and \(1.3 \%\) other ethnic groups. Suppose that a random sample of 1000 students graduating from California colleges and universities in 1998 resulted in the accompanying data on ethnic group. These data are consistent with summary statistics contained in the article titled "Crumbling Public School System a Threat to California's Future (Investor's Business Daily, November 12,1999\()\). $$ \begin{array}{lc} \text { Ethnic Group } & \text { Number in Sample } \\ \hline \text { White } & 679 \\ \text { Black } & 51 \\ \text { Hispanic } & 77 \\ \text { Asian } & 190 \\ \text { Other } & 3 \\ \hline \end{array} $$ Do the data provide evidence that the proportion of students graduating from colleges and universities in California for these ethnic group categories differs from the respective proportions in the population for California? Test the appropriate hypotheses using \(\alpha=.01\).

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