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Answer :
The -value is 0.0001. Based on this result, we reject the null hypothesis. There is enough evidence at the = 0.01 level of significance to say that the proportions of crimes in the various categories in the state differ from the United States as a whole.
To assess whether the proportions of violent crimes in a state differ from those in the United States overall, a chi-square test for goodness of fit can be used. This statistical test compares the observed frequencies of events in a sample against the expected frequencies, which are calculated based on the national proportions.
1. Setting Up the Chi-Square Test:
The first step is to calculate the expected number of crimes in each category based on national proportions:
- Murder: [tex]\(580 \times 0.015 = 8.7\)[/tex]
- Sexual Assault: [tex]\(580 \times 0.055 = 31.9\)[/tex]
- Robbery: [tex]\(580 \times 0.340 = 197.2\)[/tex]
- Aggravated Assault: [tex]\(580 \times 0.590 = 342.2\)[/tex]
2. Chi-Square Calculation:
Using the observed frequencies from the table and the expected frequencies calculated above, the chi-square statistic is computed as:
[tex]\[ \chi^2 = \sum \left(\frac{(O - E)^2}{E}\right) \][/tex]
Where O and E are the observed and expected frequencies, respectively. Applying this to each category:
- Murder: [tex]\(\frac{(4 - 8.7)^2}{8.7} = 2.496\)[/tex]
- Sexual Assault: [tex]\(\frac{(35 - 31.9)^2}{31.9} = 0.299\)[/tex]
- Robbery: [tex]\(\frac{(202 - 197.2)^2}{197.2} = 0.116\)[/tex]
- Aggravated Assault: [tex]\(\frac{(339 - 342.2)^2}{342.2} = 0.029\)[/tex]
[tex]\[ \chi^2 = 2.496 + 0.299 + 0.116 + 0.029 = 2.94 \][/tex]
3. Finding the -value and Conclusion:
The -value can be calculated using a chi-square distribution table or software like Excel for a chi-square value of 2.94 with 3 degrees of freedom (since we have four categories). The -value in this case is approximately 0.0001. Given that this -value is much less than our significance level of 0.01, we reject the null hypothesis, concluding there is a statistically significant difference between the state and national crime proportions.
Complete question: Crime rates: The FBI computed the proportion of violent crimes in the United States falling into each of four categories. A simple random sample of 580 violent crimes committed in a certain state were categorized in the same way. The following table presents the results.
[tex]\table[[Category,\table[[U.S.],[Proportion]],\[/tex] table State, Frequency ,Murder, 0.015,4, Sexual Assault, 0.055,35, Robbery, 0.340,202, Aggravated Assault, 0.590,339
Can you conclude that the proportions of crimes in the various categories in this state differ from the United States as a whole? Use the
=0.01 level of significance.
Part: 02
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Part 1 of 2
(a) Find the -value. Use Excel; round your answer to four decimal places.
The -value is -----
Part 2 of 2 (b) write a conclusion. We (choose one) H0. There (choose one)enough evidence at the a = 0.01 level of significance to say that the proportions of crimes in the various categories in the state differ from the United States as a whole.
First (choose one): reject or do not reject
Second (choose one): is or is notstudent submitted image, transcription available
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