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An article about the California lottery gave the following information on the age distribution of adults in California: [tex]$35 \%$[/tex] are between 18 and 34 years old, [tex]$51 \%$[/tex] are between 35 and 64 years old, and [tex]$14 \%$[/tex] are 65 years old and over. 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. Suppose that the data consist 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 test with [tex]$a=0.05$[/tex].

\begin{tabular}{|l|c|}
\hline Age of Purchaser & Frequency \\
\hline [tex]$18-34$[/tex] & 40 \\
\hline [tex]$35-64$[/tex] & 133 \\
\hline 65 and over & 27 \\
\hline
\end{tabular}

Calculate the test statistic. (Round your answer to two decimal places.)
[tex]\[ \chi^2 = \square \][/tex]

Use technology to calculate the P-value. (Round your answer to four decimal places.)
[tex]\[ \text{P-value} = \square \][/tex]

What can you conclude?
The data provide strong evidence to conclude that one or more of the three age groups buys a disproportionate share of lottery tickets.



Answer :

GinnyAnswer
Certainly! Let's go through the solution step-by-step:

### Step 1: Define the Hypotheses

- Null Hypothesis (H₀): The age distribution of lottery ticket purchasers follows the same distribution as the age distribution of adults in California.
- Alternative Hypothesis (H₁): The age distribution of lottery ticket purchasers is different from the age distribution of adults in California.

### Step 2: Gather the Observed Frequencies
The observed frequencies from the sample data are:

- Age 18-34: 40
- Age 35-64: 133
- Age 65 and over: 27

### Step 3: Calculate the Total Number of Observations
Sum the number of lottery ticket purchasers:

[tex]\[ \text{Total} = 40 + 133 + 27 = 200 \][/tex]

### Step 4: Determine the Expected Frequencies
The expected frequencies are calculated using the percentages given for the age distribution in California:

- Age 18-34: [tex]\( 0.35 \times 200 = 70 \)[/tex]
- Age 35-64: [tex]\( 0.51 \times 200 = 102 \)[/tex]
- Age 65 and over: [tex]\( 0.14 \times 200 = 28 \)[/tex]

These values represent what we would expect if the distribution of ages among lottery ticket purchasers matches that of the general population.

### Step 5: Perform the Chi-Square Test
To find the chi-square test statistic, we use the formula:

[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]

where [tex]\( O_i \)[/tex] is the observed frequency, and [tex]\( E_i \)[/tex] is the expected frequency.

Calculate each term:

For Age 18-34:
[tex]\[ \frac{(40 - 70)^2}{70} = \frac{900}{70} = 12.86 \][/tex]

For Age 35-64:
[tex]\[ \frac{(133 - 102)^2}{102} = \frac{961}{102} = 9.42 \][/tex]

For Age 65 and over:
[tex]\[ \frac{(27 - 28)^2}{28} = \frac{1}{28} = 0.04 \][/tex]

Now sum these values to get the test statistic:

[tex]\[ \chi^2 \approx 12.86 + 9.42 + 0.04 = 22.31 \][/tex]

### Step 6: Determine the P-value
Using a chi-square distribution with 2 degrees of freedom (since we have 3 categories and the degrees of freedom are [tex]\( n - 1 \)[/tex]), we look up the p-value corresponding to the test statistic [tex]\( \chi^2 = 22.31 \)[/tex].

We find that:

[tex]\[ \text{P-value} \approx 0.0000 \][/tex]

### Step 7: Make a Conclusion
Given that the p-value is significantly less than the significance level ([tex]\( \alpha = 0.05 \)[/tex]), we reject the null hypothesis.

Conclusion:
The data provide strong evidence to conclude that one or more of the three age groups buy a disproportionate share of lottery tickets. Thus, the age distribution of lottery ticket purchasers is significantly different from the age distribution of adults in California.