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# QUESTION l. An investigation has been carried out in a region to know the perception of &quot;citizen insecurity&quot; of its inhabitants. 1,270 people in the region were interviewed, of which 27.1% responded that it was a &quot;serious&quot; problem. Knowing that this opinion was previously held by 25.3% of the population of that region, we want to know if said opinion has changed significantly for a confidence level of 97.2%. Taking this statement into account, the following is requested: a) Critical value of the contrast statistic. b) Solve the hypothesis test and indicate what conclusion we can reach. c) P-value of contrast.

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## Answer to a math question QUESTION l. An investigation has been carried out in a region to know the perception of &quot;citizen insecurity&quot; of its inhabitants. 1,270 people in the region were interviewed, of which 27.1% responded that it was a &quot;serious&quot; problem. Knowing that this opinion was previously held by 25.3% of the population of that region, we want to know if said opinion has changed significantly for a confidence level of 97.2%. Taking this statement into account, the following is requested: a) Critical value of the contrast statistic. b) Solve the hypothesis test and indicate what conclusion we can reach. c) P-value of contrast.

Bud
4.6
To test whether the opinion about "citizen insecurity" has changed significantly in the region, you can perform a hypothesis test. Let's set up the hypothesis test and calculate the critical value, conduct the test, and find the p-value. **Hypotheses:** - Null Hypothesis $H0$: The proportion of people who consider "citizen insecurity" a "serious" problem remains the same as before, i.e., p = 0.253 $no change$. - Alternative Hypothesis $Ha$: The proportion of people who consider "citizen insecurity" a "serious" problem has changed significantly, i.e., p ≠ 0.253. **Given Data:** - Sample size $n$ = 1,270 - Proportion from the sample $p̂$ = 27.1% or 0.271 - Proportion before $p$ = 25.3% or 0.253 - Confidence level = 97.2% **a) Critical Value of the Contrast Statistic:** To find the critical value for the two-tailed test at a 97.2% confidence level, we'll use a Z-table or a calculator. The critical values for a two-tailed test at this confidence level are approximately ±2.64 $you can find this value from a standard normal distribution table or calculator$. **b) Hypothesis Test:** We'll perform a two-tailed Z-test using the given data: 1. Calculate the standard error: Standard Error $SE$ = sqrt[$p(1-p$)/n] SE = sqrt[$0.253 * 0.747$ / 1270] SE ≈ 0.0127 2. Calculate the Z-test statistic: Z = $p̂ - p$ / SE Z = $0.271 - 0.253$ / 0.0127 Z ≈ 1.417 3. Compare the Z-test statistic to the critical value: Since it's a two-tailed test, we compare the absolute value of Z to the critical value. |1.417| < 2.64 **Conclusion:** The absolute Z-test statistic $|Z|$ is less than the critical value $2.64$. Therefore, we fail to reject the null hypothesis $H0$. This means that there is no significant evidence to conclude that the proportion of people who consider "citizen insecurity" a "serious" problem has changed significantly in the region. **c) P-Value of the Contrast:** To find the p-value, you can use a standard normal distribution table or calculator. The p-value is the probability of observing a test statistic as extreme as the one calculated $Z ≈ 1.417$ under the null hypothesis. For Z ≈ 1.417, the two-tailed p-value is approximately 0.156 $from a standard normal distribution table$. Since this p-value is greater than the typical significance level $alpha$, which is usually set at 0.05, it also supports the conclusion of failing to reject the null hypothesis. There is no strong evidence of a significant change in the perception of "citizen insecurity" in the region.
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