Question

QUESTION l. An investigation has been carried out in a region to know the perception of "citizen insecurity" of its inhabitants. 1,270 people in the region were interviewed, of which 27.1% responded that it was a "serious" 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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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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