Question

Q3. Z Distribution: From Probabilities to Proportions (percentages) Answer the following questions with this scenario: To assist students with deficient math skills, the local school district decides to use the standardized math test scores to identify students who are scoring in the lowest 4% in the whole population of fifth-graders. These students will then be enrolled in after-school tutoring sessions provided by the district to improve their math skills. Helpful tips: Review the tutorial videos on “how to use the Z table” before answering the following questions. a. What is the probability of randomly drawing a student from the whole population of fifth-graders that would meet the selection criterion for the tutoring program? Explain how you determine the answer. b. If the district has 500 fifth-graders in total, how many will be identified as meeting the selection criterion for the tutoring program? c. If we use a Z distribution to represent all the test scores from the fifth-graders, what is the Z score that serves as the critical (or cutoff) value for this “significant” portion of students who will receive tutoring? Explain how you find the answer. d. This standardized math test has a published mean (μ) of 100 and standard deviation (σ) of 20 in standardized scores. Based on the critical Z score (from question c above, what would be the critical standardized score if we use the standardized scores to select students for the tutoring program? e. Taylor has a standardized score of 60, will she be enrolled in the tutoring program? Explain your rationale for this answer.

Answer 1

a) 0.04 as we're identifying those students who are in the lowest 4% of the population

b) 500* 0.04 = 20

c) The Z-score is -1.7507, This can be found using the 0.04 quantile of the standard normal in the table.

d) The critical value can be found as

e) Yes, as 60 is less than the critical value.