University Of Maryland University College Concerns

Question

University Of Maryland University College Is Concerned That Out Of Sta University of Maryland University College is concerned that out-of-state students may be receiving lower grades than Maryland students. Two independent random samples have been selected: 165 observations from population 1 (out-of-state students) and 177 from population 2 (Maryland students). The sample means obtained are X1(bar)=86 and X2(bar)=87. It is known from previous studies that the population variances are 8.1 and 7.3 respectively. Using a level of significance of 0.01, is there evidence that the out-of-state students may be receiving lower grades? Fully explain your answer.

Dr. Jack HW Helper · Accepted Answer

The concern raised by the University of Maryland University College pertains to whether out-of-state students are receiving lower academic grades compared to in-state Maryland students. To evaluate this, a hypothesis test for the difference between two population means with known variances is appropriate. The key objective is to determine if the data provide sufficient evidence at the 0.01 significance level to support the claim that out-of-state students have lower mean grades. Formulating the Hypotheses Given the context, the hypotheses are set as follows: - Null hypothesis (H0): The mean grades of out-of-state students are equal to or greater than those of Maryland students, i.e., μ1 ≥ μ2. - Alternative hypothesis (H1): The mean grades of out-of-state students are lower, i.e., μ1 Expressed mathematically: $$ H_0: \mu_1 \geq \mu_2 $$ $$ H_1: \mu_1 This is a one-tailed test focusing on detecting if out-of-state students' grades are significantly lower. Sample Data and Parameters - Sample size for out-of-state students: n1 = 165 - Sample mean for out-of-state students: X1̄ = 86 - Population variance for out-of-state students: σ1² = 8.1 - Sample size for Maryland students: n2 = 177 - Sample mean for Maryland students: X2̄ = 87 - Population variance for Maryland students: σ2² = 7.3 From these, the known population standard deviations are: \[ \sigma_1 = \sqrt{8.1} \approx 2.847 $$ $$ \sigma_2 = \sqrt{7.3} \approx 2.703 $$ Methodology Given the known population variances, a z-test for the difference between two means is suitable. The test statistic is calculated as: $$ Z = \frac{(X_1̄ - X_2̄)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} $$ Substituting the known values: $$ Z = \frac{(86 - 87)}{\sqrt{\frac{8.1}{165} + \frac{7.3}{177}}} $$ Calculating the denominator: $$ \frac{8.1}{165} \approx 0.0491 $$ $$ \frac{7.3}{177} \approx 0.0412 $$ $$ \text{Sum} = 0.0903 $$ $$ \sqrt{0.0903} \approx 0.3005 $$ Calculating the numerator: $$ 86 - 87 = -1 $$ Thus, the test

University Of Maryland University College Is Concerned That Out Of Sta

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