Random Sample Of 900 People | Educational Q&A

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Perform a hypothesis test to determine if the proportion of people in the UK who agree with a statement about the country's future economy is different from that of the US, based on the given sample data. Additionally, calculate the p-value. Also, perform a variance hypothesis test comparing two samples with given variances and sample sizes at a 5% significance level.

Paper For Above instruction

The purpose of this paper is to analyze two different statistical hypotheses based on survey data collected from samples in the United Kingdom and the United States. The primary focus is to assess whether the proportions of individuals who agree with positive statements about the country's economic future differ significantly between these two nations. Additionally, it examines a variance comparison between two samples to determine if there is statistical evidence of a variance difference. These analyses involve hypothesis testing at specified significance levels with the calculation of p-values to interpret the results.

Part 1: Comparing Proportions of UK and US Respondents

In the first scenario, a sample of 900 people from the UK showed that 594 agreed with the statement "the country's future economy is positive." Correspondingly, in the US, 540 of 900 people agreed. We are tasked with testing whether the proportion of agreement differs between these two groups at a 1% significance level.

First, identify the hypotheses:

• Null hypothesis (H₀): p₁ = p₂ (The proportion of agreement is the same in both the UK and US)
• Alternative hypothesis (H₁): p₁ ≠ p₂ (The proportion of agreement differs between the UK and US)

Calculate the sample proportions:

p̂₁ = 594 / 900 = 0.66

p̂₂ = 540 / 900 = 0.60

Next, compute the pooled proportion (p̂):

p̂ = (x₁ + x₂) / (n₁ + n₂) = (594 + 540) / (900 + 900) = 1134 / 1800 = 0.63

Calculate the standard error (SE):

SE = √[ p̂ (1 - p̂) (1/n₁ + 1/n₂) ] = √[ 0.63 0.37 (1/900 + 1/900) ]

SE ≈ √[ 0.2331 * 2/900 ] ≈ √(0.000517) ≈ 0.0227

Compute the z-score:

z = (p̂₁ - p̂₂) / SE = (0.66 - 0.60) / 0.0227 ≈ 0.06 / 0.0227 ≈ 2.639

At a 1% significance level (α = 0.01), the critical z-values for a two-tailed test are approximately ±2.576.

Since 2.639 > 2.576, we reject the null hypothesis, indicating a significant difference in proportions. The data suggest that the proportion of people in the UK who agree with the statement is statistically different from that of the US.

Part 2: p-Value Calculation

The p-value corresponds to the probability of observing a z-score as extreme as 2.639 under the null hypothesis. Using standard normal distribution tables or software, the p-value for a two-tailed test is:

p-value = 2 P(Z > 2.639) ≈ 2 0.0042 ≈ 0.0084

Since the p-value (≈0.0084) is less than the significance level (0.01), it reinforces the conclusion to reject the null hypothesis, confirming significant difference.

Part 3: Variance Hypothesis Test

In the second scenario, two samples are given with sample sizes n₁ = 13 and n₂ = 21, and variances s₁² = 1450 and s₂² = 1320. The hypotheses to test whether variances differ are:

• Null hypothesis (H₀): σ₁² ≤ σ₂² (or equivalently, variance in sample 1 is less than or equal to sample 2)
• Alternative hypothesis (H₁): σ₁² > σ₂² (variance in sample 1 is greater than sample 2)

This is a one-sided F-test for variances. The test statistic is:

F = s₁² / s₂² = 1450 / 1320 ≈ 1.098

Degrees of freedom are:

• df₁ = n₁ - 1 = 12
• df₂ = n₂ - 1 = 20

Using F-distribution tables or software, we determine the critical value at a 5% significance level for df₁ = 12 and df₂ = 20. The critical F-value (one-tailed) is approximately 2.50.

Since the calculated F-value (1.098) is less than 2.50, we fail to reject the null hypothesis. This indicates that there isn't sufficient evidence to conclude that the variance in sample 1 exceeds that in sample 2.

Conclusion

The analysis of proportions from UK and US samples reveals a statistically significant difference in agreement levels about the economic outlook, with the UK showing a higher proportion. The p-value confirms the significance of this difference. Conversely, the variance comparison indicates no statistically significant difference between the samples at the 5% level. These findings provide insights into perceptions of economic optimism and the variability of survey responses across different samples.

References

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