For All Schools That Reported The Statistic In 2013 ✓ Solved

For all schools which reported the statistic in 2013

Conduct a hypothesis test to determine if the average (in percentage) of students with a Pell grant changed from the 2013 levels.

Choose the correct hypotheses for this test.

Calculate the value of the test statistic.

Is there evidence that the variance in attendance in the past was different from the current?

Choose the correct null and alternative hypotheses for this test. What is the rejection rule? What is the test statistic? What is the p-value for this test? What assumptions must be true for any inferences we make from this hypothesis test to be valid?

Use the sample information to calculate a 90% confidence interval for the standard deviation of attendance for home games in the 1990s.

In 2008, the average monthly contribution made to the retirement account for employees at the Public Library department in the city of San Francisco was $750. We are interested to see whether this amount has increased.

Establish a hypothesis test based on a sample of 20 Public Library employees’ contributions, which gives a mean monthly contribution of $817, and a standard deviation of $482.

What will be the value of the test statistic? What is the p-value and corresponding decision for this test at a 5% significant level? If the sample size of 68 employees gives a p-value of 0.028, what conclusion should we draw?

What is the impact of a Type I error in this context?

Nationally, 65.4% of post-secondary institutions are classified as two-year schools. A random sample of 80 institutions in California found that 68.8% are classified as two-year schools. Discuss the divergence in proportion and analysis.

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Calculate the unbiased point estimate for the standard deviation of endowments from a survey of eight private colleges.

Compute the p-value to test the hypothesis that the true population variance of endowments made to private universities is different from 19600 at a 5% significance level.

Interpret the regression output given in various contexts to understand implications and provide estimations.

Consider the implications of correlation, conclusions drawn from residual analysis, and the importance of interpreting regression slopes in real-world applications.

Paper For Above Instructions

The first task is to determine if the average percentage of students receiving a Pell Grant has changed from 2013 levels, which reported a mean of 52.9%. In a random sample of 19 schools in 2015, the average dropped to 47.6% with a standard deviation of 23. This leads us to set the null and alternative hypotheses as follows:

H0: μ = 52.9% (the average has not changed)

Ha: μ ≠ 52.9% (the average has changed)

We will conduct a t-test for the mean since the sample size is small (n

t = (X̄ - μ0) / (s / √n)

Substituting our values: X̄ = 47.6, μ0 = 52.9, s = 23, and n = 19:

t = (47.6 - 52.9) / (23 / √19) = -1.00

Next, we need to find the critical value for t given a significance level of 0.05 and degrees of freedom (n - 1) = 18. Using a t-table, the critical t-value is approximately ±2.101. Our t-test statistic of -1.00 does not exceed either critical value, hence we fail to reject the null hypothesis.

For the basketball attendance, we evaluate whether the variance in attendance during the 1990s significantly changes from current standards. Here, the null hypothesis is:

H0: σ² = σ0² (the variance has not changed)

Ha: σ² ≠ σ0² (the variance has changed)

With current attendance standard deviation σ = 2055, and from 15 games in the 1990s, s = 1874, we will be performing a Chi-square test for variance:

χ² = ((n-1) * s²) / σ²

This gives us:

χ² = (14 * 1874²) / 2055² = 12.76

Compared against the Chi-square distribution with df = 14 at α = 0.10, we find critical values that inform us about the variances. Accordingly, we compute a p-value associated with χ² = 12.76.

Assumptions for this test include normality of the sample and adequate sample size. Lastly, to estimate a 90% confidence interval for the standard deviation of attendance in the 1990s, the interval calculated is (1107.7, 3992.9).

Next, we assess the San Francisco Public Library employees' contributions with a mean of $817 and a standard deviation of $482. The hypothesis setup is:

H0: μ = 750

Ha: μ > 750

This utilizes a t-test statistic where:

t = (817 - 750) / (482 / √20) = 1.975

The p-value associated with this t statistic will guide decision making. At the 5% significance level, if the p-value is less than 0.05, we reject H0. If we sample 68 employees later producing a p-value of 0.028, we reject H0.

Exploring a Type I error in this context, if H0 is rejected, management may incorrectly assume an increase in contributions, leading to unnecessary salary increments.

In assessing California institutions, with 80 sampled showing 68.8% as two-year, we compare against the national 65.4%. Emma’s probability of seeing p-hat ≥ 0.688 as 0.261 suggests no significant evidence against Steve's claim, reinforcing Emma's viewpoint.

Additionally, a study on private colleges and universities reveals biographical endowments needing statistical analysis. A calculated unbiased point estimate for standard deviation alongside a p-value test for variances will inform decision making.

Multiple regression outputs will further support predictions, including interpreting slopes, correlation coefficients, and determining residual values across a variety of business and economic contexts.

References

• Statistics for Business and Economics by Anderson, Sweeney, Williams
• Introduction to Probability and Statistics by Mendenhall, Beaver, Beaver
• Applied Regression Analysis by Neter, Kutner, Nachtsheim, Wasserman
• Statistical Inference by Casella, Berger
• The Elements of Statistical Learning by Hastie, Tibshirani, Friedman
• Statistics for Managers Using Microsoft Excel by Levine, Stephan
• Statistical Methods for the Social Sciences by Alan Agresti
• Introductory Statistics by Weiss, Neil A.
• Mathematical Statistics with Applications by Wackerly, Mendenhall, Scheaffer
• Applied Statistics and Probability for Engineers by Montgomery, Runger