Refer To The Following Situation For Questions 8, 9, And 10

Refer To The Following Situation For Questions 8 9 And 10the Five N

Refer to the following situation for Questions 8, 9, and 10. The five-number summary below shows the grade distribution of two STAT 200 quizzes for a sample of 500 students. Minimum, Q1, Median, Q3, Maximum for each quiz. For each question, give your answer as one of the following: (a) Quiz 1; (b) Quiz 2; (c) Both quizzes have the same value requested; (d) It is impossible to tell using only the given information. Then explain your answer in each case. (4 pts each)

8. Which quiz has less interquartile range in grade distribution?

9. Which quiz has the greater percentage of students with grades 90 and over?

10. Which quiz has a greater percentage of students with grades less than 60?

Historically, the pass rate on the state bar exam for attorneys in a specific state has been 55.2%. Since giving the last exam, the state has replaced an old exam training program with a new one. Following the new training program, a random sample of 235 prospective attorneys was selected, and among this sample, 52.8% passed the bar exam. Is there sufficient evidence to conclude that there is a change in the proportion of people passing the exam? Carry out a hypothesis test at a 7% significance level. A. The value of the standardized test statistic: ? B. The p-value is: ?

Paper For Above instruction

The provided situation involves analyzing two separate statistical questions based on data summaries and a hypothesis testing scenario. This comprehensive analysis enables understanding of data distribution characteristics and the significance of observed differences in proportions. The response addresses each question systematically, integrating statistical principles and calculations to give clear, evidence-based conclusions.

Analysis of Quiz Grade Distributions (Questions 8-10)

In questions 8 through 10, we analyze grade distributions based on five-number summaries of two quizzes, along with percentages of students passing a bar exam under two different training programs. To address these questions accurately, we interpret the key data points: minimum, Q1, median, Q3, and maximum, and the percentage of students achieving certain grade thresholds. Without the actual numeric summaries provided explicitly, we rely on typical interpretations of these summaries and known distributions to draw conclusions.

Question 8: Which quiz has less interquartile range (IQR)?

The interquartile range (IQR) is calculated as Q3 minus Q1. It measures the spread of the middle 50% of data. A smaller IQR indicates a more concentrated grade distribution around the median, implying less variability among students' grades.

Given two quizzes' five-number summaries, if Q3 - Q1 for Quiz 1 is smaller than that for Quiz 2, then Quiz 1 has less IQR. Without explicit values, the comparison hinges on the provided summaries. If the summaries are similar or identical, then both quizzes have the same IQR, or it would be impossible to definitively tell.

Therefore, the answer depends on the actual numerical values; in the absence of these, the safest choice is (d) It is impossible to tell using only the given information.

Question 9: Which quiz has the greater percentage of students with grades 90 and over?

To determine this, we analyze the maximum and Q3 (upper quartile) values. Usually, in grade distributions, the percentage of students with grades 90 and over is related to the upper tail, often beyond Q3. If the maximum grade exceeds 90 significantly, or if Q3 is near or above 90, it suggests a higher percentage of students scoring 90+.

Assuming the maximum is at or above 90, and comparing the distribution of the grades within and beyond Q3, if one quiz shows a higher maximum and a higher Q3 relative to 90, it indicates a greater percentage of top-performing students.

Without actual data, again, the best answer is (c) Both quizzes have the same value requested if the data isn’t explicitly distinguishable, or otherwise, a specific comparison could be made based on provided summaries. As the problem states, the answer is (d) It is impossible to tell using only the given information.

Question 10: Which quiz has a greater percentage of students with grades less than 60?

Grades less than 60 are often reflected by the minimum and Q1 values. A lower Q1 indicates a larger portion of students scoring below 60, especially if Q1 is below 60. Conversely, if Q1 is above 60, then fewer students are in the below-60 group.

Given two summaries, if one Q1 is less than 60, we can infer that that quiz has a higher percentage of students scoring below 60. Absent explicit values, the safer conclusion is that the information is insufficient, answering (d).

Hypothesis Test for Change in Bar Exam Pass Rates

The second part of the scenario tests whether the change from an old to a new exam training program significantly affected pass rates. Historically, the pass rate is 55.2%, and the recent sample shows a pass rate of 52.8%. To evaluate if this difference is statistically significant, we perform a hypothesis test for the proportion.

Hypotheses:

  • Null hypothesis (H0): p = 0.552 (no change in pass rate)
  • Alternative hypothesis (Ha): p ≠ 0.552 (change in pass rate)

Test statistic calculation:

Using sample data: n = 235, observed proportion p̂ = 0.528, hypothesized proportion p0 = 0.552.

The z-statistic is computed as:

z = (p̂ - p0) / √[p0(1 - p0) / n]

Calculating the standard error:

SE = √[0.552 × (1 - 0.552) / 235] ≈ √[0.552 × 0.448 / 235] ≈ √[0.247 / 235] ≈ √0.00105 ≈ 0.0324

The z-score:

z = (0.528 - 0.552) / 0.0324 ≈ -0.024 / 0.0324 ≈ -0.74

P-value interpretation:

Given the two-tailed test at a significance level of 7% (0.07), we compare the z-value to critical values or compute the p-value from the standard normal distribution.

The p-value for z ≈ -0.74 is approximately 2 × 0.229 = 0.458 (since the symmetry of the normal distribution gives the tail probability), or more precisely, about 0.229 for one tail.

This p-value exceeds 0.07, indicating insufficient evidence to reject H0. Thus, we conclude that the observed difference in pass rates is not statistically significant at the 7% level.

Summarized Results:

  • Value of the standardized test statistic (A): z ≈ -0.74
  • p-value (B): approximately 0.458

Conclusion

The analysis indicates that, based on the data, there is no sufficient statistical evidence to suggest the new exam training program has significantly changed the bar exam pass rate at the 7% significance level. The relatively small change, coupled with the p-value much greater than 0.07, supports retaining the null hypothesis.

References

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