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Sagot :
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.
### 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.
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