Population Means With Unknown Standard Deviations
Population Means With Unknown Standard Deviationsthese Are To Be Comp
Population Means with unknown standard deviations. These are to be complete Hypothesis test starting with Ho & Ha and ending with a decision based on the rejection region or the p-value. #78 The mean number of English courses taken in a two–year time period by male and female college students is believed to be about the same. An experiment is conducted and data are collected from 29 males and 16 females. The males took an average of three English courses with a standard deviation of 0.8. The females took an average of four English courses with a standard deviation of 1.0. Are the means statistically the same? #79 A student at a four-year college claims that mean enrollment at four–year colleges is higher than at two–year colleges in the United States. Two surveys are conducted. Of the 35 two–year colleges surveyed, the mean enrollment was 5,068 with a standard deviation of 4,777. Of the 35 four-year colleges surveyed, the mean enrollment was 5,466 with a standard deviation of 8,191
Paper For Above instruction
Introduction
The comparison of population means when the standard deviations are unknown is a common problem in inferential statistics. Such scenarios necessitate the use of t-tests for independent samples, which accommodate the uncertainty inherent in estimating standard deviations from sample data. This paper addresses two specific problems involving independent samples with unknown standard deviations: first, whether male and female college students take an equivalent number of English courses; second, whether the mean enrollment at four-year colleges exceeds that of two-year colleges. Both problems involve formulating hypotheses, calculating test statistics, determining rejection regions or p-values, and making informed conclusions based on the data.
Problem 1: Comparing Means of English Courses Taken by Males and Females
Research Question: Are the mean number of English courses taken in a two-year period statistically the same for male and female students?
Hypotheses:
- Null hypothesis (H₀): μ₁ = μ₂ (The mean number of English courses is the same for males and females)
- Alternative hypothesis (H₁): μ₁ ≠ μ₂ (The means are different)
Data Summary:
- Males: sample size n₁=29, mean \(\bar{x}_1=3.0\), standard deviation s₁=0.8
- Females: sample size n₂=16, mean \(\bar{x}_2=4.0\), standard deviation s₂=1.0
Test Statistic Calculation:
Using the two-sample t-test for independent samples with unequal variances (Welch's t-test):
\[
t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
\]
\[
t = \frac{3.0 - 4.0}{\sqrt{\frac{(0.8)^2}{29} + \frac{(1.0)^2}{16}}} = \frac{-1.0}{\sqrt{\frac{0.64}{29} + \frac{1.0}{16}}}
\]
\[
t = \frac{-1.0}{\sqrt{0.02207 + 0.0625}} = \frac{-1.0}{\sqrt{0.08457}} = \frac{-1.0}{0.2909} \approx -3.44
\]
Degrees of Freedom:
Calculated using the Welch–Satterthwaite equation:
\[
df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}
\]
Plugging values:
\[
df \approx \frac{(0.02207 + 0.0625)^2}{\frac{(0.02207)^2}{28} + \frac{(0.0625)^2}{15}} \approx \frac{0.00716}{0.0000174 + 0.000260} \approx 26.7
\]
rounded to 27 degrees of freedom.
Decision:
Using a two-tailed test at significance level \(\alpha=0.05\), the critical t-value is approximately ±2.052. Since |t| ≈ 3.44 > 2.052, we reject H₀.
Conclusion:
There is statistically significant evidence at the 5% level to conclude that male and female students differ in the number of English courses taken in a two-year period.
Problem 2: Comparing College Enrollment Sizes
Research Question: Is the mean enrollment at four-year colleges higher than at two-year colleges?
Hypotheses:
- Null hypothesis (H₀): μ₁ ≥ μ₂ (Average enrollment at four-year colleges is less than or equal to that at two-year colleges)
- Alternative hypothesis (H₁): μ₁
Data Summary:
- Two-year colleges: n₁=35, mean = 5,068, s₁=4,777
- Four-year colleges: n₂=35, mean = 5,466, s₂=8,191
Test Statistic Calculation:
Using the two-sample t-test for independent samples with unequal variances:
\[
t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
\]
\[
t = \frac{5,068 - 5,466}{\sqrt{\frac{4,777^2}{35} + \frac{8,191^2}{35}}}
\]
Calculating numerator:
\[
\bar{x}_1 - \bar{x}_2 = -398
\]
Calculating denominator:
\[
\frac{4,777^2}{35} = \frac{22,813,729}{35} \approx 651,823
\]
\[
\frac{8,191^2}{35} = \frac{67,038,481}{35} \approx 1,915,097
\]
Sum:
\[
651,823 + 1,915,097 = 2,566,920
\]
Square root:
\[
\sqrt{2,566,920} \approx 1,603.4
\]
Test statistic:
\[
t = \frac{-398}{1,603.4} \approx -0.248
\]
Degrees of Freedom:
Using Welch–Satterthwaite approximation:
\[
df \approx \frac{(651,823 + 1,915,097)^2}{\frac{(651,823)^2}{34} + \frac{(1,915,097)^2}{34}}
\]
Due to the large variances, the degrees of freedom are approximately less than 70, but for simplicity, assume df ≈ 34.
Decision:
Since the test is one-tailed with H₀: μ₁ ≥ μ₂, and the calculated t ≈ -0.248 is close to zero, and the critical t-value at \(\alpha=0.05\) for df=34 is approximately -1.690.
Because -0.248 > -1.690, we fail to reject H₀.
Conclusion:
There is no statistically significant evidence to support the claim that the mean enrollment at four-year colleges exceeds that at two-year colleges at the 5% significance level.
Discussion
The first analysis confirms a significant difference in the number of English courses undertaken by males and females, with females taking more courses on average. This could reflect differing academic interests or course load preferences, which merit further investigation. The second analysis indicates that the claim of higher enrollment at four-year colleges is not supported by the data; the observed difference is not statistically significant, possibly due to high variability and sample size constraints.
Understanding the statistical methods applied—namely, the Welch’s t-test—is critical when standard deviations are unknown and unequal. These tests are flexible for real-world data, accommodating unequal variances and different sample sizes, which often occur in social science research.
Conclusion
The application of hypothesis testing in these contexts provides valuable insights into educational data. The methodology underscores the importance of accurate assumptions, correct calculation of test statistics, and interpretation within the context of the chosen significance level. Policymakers and educators can utilize such analyses to inform decisions based on empirical evidence.
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