It has been reported that 10.3% of U.S. households do not own a vehicle, with 34.2% owning 2 vehicles, and 17.1% owning 3 or more vehicles. The data for a random sample of 100 households in a resort community are summarized in the frequency distribution below. At the 0.05 level of significance, can we reject the possibility that the vehicle-ownership distribution in this community differs from that of the nation as a whole?
Number of Numberof
Vehicles Owned Households
0 20
1 35
2 23
3 or more 22
100
100
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To determine if we can reject the possibility that the vehicle-ownership distribution in the resort community differs from that of the nation as a whole, we can perform a chi-square goodness-of-fit test.
Step 1: Set up the hypotheses:
- Null hypothesis (H0): The vehicle-ownership distribution in the resort community is the same as that of the nation as a whole.
- Alternative hypothesis (H1): The vehicle-ownership distribution in the resort community differs from that of the nation as a whole.
Step 2: Choose the significance level (α):
The significance level represents the probability of rejecting the null hypothesis when it is true. In this case, the significance level is given as 0.05.
Step 3: Calculate the expected frequencies:
Calculate the expected frequencies for each category under the assumption that the vehicle-ownership distribution in the resort community is the same as that of the nation as a whole. The expected frequency for each category can be calculated by multiplying the proportion of households in the nation by the total number of households in the resort community.
For example, for category 0 (no vehicles owned):
Expected frequency = Proportion in nation * Total number of households in the resort community
= 0.103 * 100
= 10.3
Similarly, calculate the expected frequencies for other categories:
Category 1: 0.103 * 100 = 10.3
Category 2: 0.342 * 100 = 34.2
Category 3 or more: 0.171 * 100 = 17.1
Step 4: Perform the chi-square test:
Using the observed frequencies (given in the frequency distribution) and the expected frequencies, calculate the chi-square test statistic.
Chi-square test statistic (χ²) = Σ [(observed frequency - expected frequency)² / expected frequency]
Sum up the values for all categories.
Step 5: Determine the critical value:
Find the critical value from the chi-square distribution table with degrees of freedom equal to the number of categories minus 1. In this case, there are 4 categories, so the degrees of freedom is 3. Look up the critical value at the significance level of 0.05.
Step 6: Compare the test statistic with the critical value:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 7: Make a decision:
If the null hypothesis is rejected, we can conclude that there is evidence to suggest that the vehicle-ownership distribution in the resort community differs from that of the nation as a whole. If the null hypothesis is not rejected, we do not have enough evidence to suggest a difference in the distributions.
Performing the calculations using the given data will yield the test statistic value, which can be compared with the critical value to make a decision.