Random Sample Of 900 People In The UK Were Asked To Respond
Qa Random Sample Of 900 People In The Uk Were Asked To Respond To Thi
Qa random sample of 900 people in the UK were asked to respond to this statement: “the country’s future economy is positive”. Of these sample people, 594 agreed with the statement. When the same statement was presented to a random sample of 900 people in the US, 540 agreed with the statement. a) Using a 1% significance level, perform a hypothesis test to determine if the proportion of people in the UK who agreed with the statement is different from that of the US. b) Determine the p-value. Q2: Given the null and alternative hypotheses H0: σ² ≤ σ² and H1: σ² > σ², and the following sample information: Sample 1, sample size = 13, sample variance = 1450; Sample 2, sample size = 21, sample variance = 1320. Test the hypothesis at 5% level of significance and indicate whether the null hypothesis should be rejected.
Paper For Above instruction
Introduction
Understanding comparative attitudes between populations is a key aspect of social science research, especially in gauging economic optimism through survey data. The current analysis involves two primary statistical tests: a hypothesis test for the difference in proportions to evaluate whether the UK and US populations differ significantly in their agreement with a statement about the country's economic future, and a variance test to compare the variability of two sample groups. These tests provide insights into national sentiments and variability in economic outlooks, respectively.
Hypothesis Testing for Difference in Proportions
The survey data indicates that 66% of the UK sample (594 out of 900) agreed with the statement, whereas 60% of the US sample (540 out of 900) agreed. To determine if these differences are statistically significant at the 1% significance level, we set the null hypothesis H0: p1 = p2, meaning there is no difference in population proportions, against the alternative hypothesis H1: p1 ≠ p2, indicating a difference exists.
The test employs a two-proportion z-test, which compares observed difference in sample proportions to the hypothesized difference under H0. The pooled proportion (p̂) is calculated as:
\[
p̂ = \frac{x_1 + x_2}{n_1 + n_2} = \frac{594 + 540}{900 + 900} = \frac{1134}{1800} = 0.63
\]
The standard error (SE) for the difference is:
\[
SE = \sqrt{ p̂ (1 - p̂) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } = \sqrt{ 0.63 \times 0.37 \left( \frac{1}{900} + \frac{1}{900} \right) } \approx 0.0227
\]
The observed difference in proportions:
\[
d = p_1 - p_2 = 0.66 - 0.60 = 0.06
\]
Corresponding z-statistic:
\[
z = \frac{d}{SE} = \frac{0.06}{0.0227} \approx 2.64
\]
Referring to standard normal distribution tables, the two-tailed p-value associated with z = 2.64 is approximately 0.0083.
Since p-value
Conclusion for Part A
The hypothesis test provides strong evidence at the 1% significance level to conclude that the proportion of people in the UK who agree with the optimistic statement about the economy differs from that in the US.
Finding the p-value
The computed p-value of approximately 0.0083 confirms the statistical significance of the observed difference, as it falls below the 0.01 threshold. This indicates a low probability that such a difference would be observed if the true proportions were equal, underscoring the significance of the difference.
Variance Comparison Between Two Samples
Next, the analysis assesses whether there is a significant difference in the variances of two samples, using hypotheses:
\[
H_0 : \sigma^2_1 \leq \sigma^2_2 \quad \text{vs.} \quad H_1 : \sigma^2_1 > \sigma^2_2
\]
Given:
- Sample 1: n1=13, s1²=1450
- Sample 2: n2=21, s2²=1320
The test employs the F-test for equality of variances, calculated as:
\[
F = \frac{ s^2_1 }{ s^2_2 } = \frac{ 1450 }{ 1320 } \approx 1.098
\]
Degrees of freedom:
- df1 = n1 - 1 = 12
- df2 = n2 - 1 = 20
To determine whether to reject H0 at the 5% significance level, the critical F-value (Fcritical) for a one-tailed test with df1=12 and df2=20 is approximately 2.51 (from F-distribution tables).
Since the calculated F-value (1.098) is less than Fcritical (2.51), there is insufficient evidence to reject H0; conflicting variances are not statistically significant, and the observed ratio does not indicate a difference in variability at the 5% level.
Conclusion for Variance Test
The analysis suggests that, based on the sample data, the variability in the two groups does not significantly differ at the 5% level, and H0 cannot be rejected.
Overall Implications
The results underscore meaningful differences in perceptions of economic positivity between the UK and US populations, perhaps reflecting underlying cultural or economic factors. Conversely, the variances in measured data from the samples do not significantly differ, hinting at comparable levels of variability in the measured attribute or outcome.
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