Q1 In A Random Sample Of 397 Adult US Citizens

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Analyze a series of hypothesis tests involving proportions within adult U.S. populations. The scenarios focus on determining whether sample data provides statistically significant evidence to support claims about the population proportion of U.S. adults holding a Bachelor's degree or the proportion being other than a specific value. Each problem involves formulating null and alternative hypotheses, calculating test statistics, estimating p-values, and interpreting results at given significance levels.

Paper For Above instruction

The analysis of hypotheses concerning population proportions is a fundamental aspect of inferential statistics. These tests help determine whether sample data provide sufficient evidence to support claims about population parameters. In the context of U.S. adult education and demographic studies, such tests are crucial for understanding national educational attainment and demographic characteristics. The following examination involves four distinct hypotheses testing scenarios, all revolving around proportions in adult populations, with specific focus on holding a Bachelor's degree or the prevalence of left-handedness, employing different significance levels and hypotheses for each case.

Hypothesis Testing Framework

Every hypothesis test begins with the formulation of a null hypothesis (H0) which represents no effect or status quo, and an alternative hypothesis (Ha) that reflects the research question or the effect in question. The test involves calculating a test statistic—typically a z-score for large sample proportions—and then deriving the p-value, which indicates the probability of observing a result as extreme or more extreme than the sample data under the null hypothesis.

The critical significance level (α) determines the threshold for rejecting H0. If the p-value is less than α, we reject the null hypothesis, indicating statistically significant evidence supporting the alternative. Conversely, if the p-value exceeds α, we fail to reject H0.

Scenario 1: Proportion of US Adults with Bachelor's Degree Less Than 0.42

In a sample of 397 adults aged 25 or over, 123 are observed to hold a Bachelor's degree. The null hypothesis asserts that the true proportion, p, is 0.42, while the alternative states that p is less than 0.42.

Calculating the sample proportion:

p̂ = 123 / 397 ≈ 0.309

The test statistic (z) is computed as:

z = (p̂ - p0) / √[ p0 (1 - p0) / n ]


= (0.309 - 0.42) / √[ 0.42 * 0.58 / 397 ]

This yields a z-value approximately equal to -4.54. The p-value, being the probability of observing a z-value less than -4.54, is very small (p 0.

Thus, the sample provides strong evidence that the true proportion of US adults with a Bachelor's degree is lower than 0.42.

Scenario 2: Proportion of US Adults with Bachelor's Degree Different From 0.4

In a sample of 404 adults, 120 hold a Bachelor's degree, giving:

p̂ = 120 / 404 ≈ 0.297

Null hypothesis: p = 0.4; Alternative: p ≠ 0.4.

Calculating the z-statistic:

z = (p̂ - p0) / √[ p0 (1 - p0) / n ]


= (0.297 - 0.4) / √[ 0.4 * 0.6 / 404 ]

The z-value is approximately -2.545. This corresponds to a two-tailed p-value of about 0.0109. At a significance level of α = 0.012, since p-value ≈ 0.0109 0. Therefore, there is statistically significant evidence that the true proportion differs from 0.4.

Scenario 3: Higher Than 0.04 Left-Handed Population Proportion

A single sample of 104 individuals reveals 8 left-handed persons:

p̂ = 8 / 104 ≈ 0.077

Null hypothesis: p = 0.04; Alternative: p > 0.04.

The z-statistic:

z = (p̂ - p0) / √[ p0 (1 - p0) / n ]


= (0.077 - 0.04) / √[ 0.04 * 0.96 / 104 ]

Calculated z-value is approximately 1.438. The p-value, area to the right of z=1.438, is approximately 0.075. Since α = 0.01, and p-value ≈ 0.075 > 0.01, we fail to reject H0. There is insufficient evidence to conclude that the true proportion of left-handed individuals exceeds 0.04.

Scenario 4: Proportion of Left-Handed Individuals Not Equal to 0.04

In a sample of 110 individuals, 9 are left-handed:

p̂ = 9 / 110 ≈ 0.082

Null hypothesis: p = 0.04; Alternative: p ≠ 0.04.

Calculating the z-score:

z = (p̂ - p0) / √[ p0 (1 - p0) / n ]


= (0.082 - 0.04) / √[ 0.04 * 0.96 / 110 ]

This yields a z-value of approximately 1.569. The two-tailed p-value is around 0.116. Since α = 0.008, and p-value > α, we fail to reject H0. The evidence does not support a population proportion different from 0.04.

Conclusion

Across these scenarios, hypothesis testing reveals that sample data can strongly support claims about population proportions. When p̂ significantly diverges from p0, and the p-value falls below the significance threshold, the null hypothesis is rejected. Conversely, insufficient evidence leads us to fail to reject H0. These analyses exemplify the importance of selecting appropriate significance levels and accurately computing test statistics and p-values in inferential statistics applications.

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