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Chapter 6: Problem 18

Students at a particular university use a telephone registration system to select their courses for the next term. There are four different priority groups, with students in Group 1 registering first, followed by those in Group 2, and so on. Suppose that the university provided the accompanying information on registration for the fall semester. The entries in the table represent the proportion of students falling into each of the 20 priority-unit combinations. $$\begin{array}{c|ccccc} \begin{array}{c} \text { Priority } \\ \text { Group } \end{array} & \mathbf{0 - 3} & \mathbf{4 - 6} & \mathbf{7 - 9} & \mathbf{1 0 - 1 2} & \begin{array}{c} \text { More } \\ \text { Than 12 } \end{array} \\ \hline \mathbf{1} & .01 & .01 & .06 & .10 & .07 \\ \mathbf{2} & .02 & .03 & .06 & .09 & .05 \\ \mathbf{3} & .04 & .06 & .06 & .06 & .03 \\ \mathbf{4} & .04 & .08 & .07 & .05 & .01 \\ & & & & & \end{array}$$ a. What proportion of students at this university got 10 or more units during the first call? b. Suppose that a student reports receiving 11 units during the first call. Is it more likely that he or she is in the first or the fourth priority group? c. If you are in the third priority group next term, is it likely that you will get more than 9 units during the first call? Explain.

Short Answer

Expert verified
a. The proportion of students who got 10 or more units during the first call is 0.46.\nb. It's more likely that the student who received 11 units during the first call belongs to the first priority group.\nc. It's not likely for a student in the third priority group to get more than 9 units during the first call.

Step by step solution

01

Calculate Proportion for Students with 10 or More Units

To find out the proportion of students who got 10 or more units during the first call, sum all the proportions of students in priority groups who got 10-12 units and more than 12 units. That is \(0.10+0.07+0.09+0.05+0.06+0.03+0.05+0.01=0.46\).
02

Find the Priority Group for 11 Units

If a student reports receiving 11 units during the first call, they could either belong to the 10-12 units category or the more than 12 units category. Compare the proportions of students in priority group 1 and 4 for these categories. For priority group 1 it will be \(0.10+0.07 = 0.17\) and for priority group 4 it will be \(0.05+0.01 = 0.06\). Since 0.17 is greater than 0.06, the student is more likely to be in the first priority group.
03

Calculate Likelihood for Group 3 with More Than 9 Units

To determine if it's likely for a student in the third priority group to get more than 9 units during the first call, sum the proportions of students in the third priority group who got 10-12 units and more than 12 units. That is \(0.06+0.03=0.09\). Since the proportion is less than 0.5, it is not likely that a student in the third priority group will get more than 9 units during the first call.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Calculation
Understanding the concept of proportion calculation is key to many statistical analyses, including university course registration outcomes. In essence, proportion refers to the part of a whole expressed as a fraction or percentage. When calculating the proportion of students getting 10 or more units in our exercise, we consider the sum of the respective categories across all groups. Here鈥檚 a simplified breakdown:

To calculate this, you would add the proportions for each priority group who fell into the 10-12 units and more than 12 units categories. For instance, priority group 1 had 0.10 in the 10-12 unit range and 0.07 in the more than 12 unit range, contributing a total of 0.17 to the final sum. Repeating this process for all groups and then summing their contributions, you get a total proportion of 0.46. This means 46% of students were able to secure 10 or more units during the first call.
Priority Group Analysis
Priority group analysis is a process used to determine the characteristics or outcomes of different categorized groups within a population. In the context of the course registration statistics, it helps to analyze the likelihood of students obtaining a certain number of units based on their priority group.

In order to determine whether a student receiving 11 units is more likely from Group 1 or Group 4, we analyze the sum of the proportions in categories relevant to 11 units (10-12, more than 12). As explained in the solution steps, Group 1 has a total of 0.17 while Group 4 has only 0.06, indicating that it is more probable for a student with 11 units to belong to Group 1. This type of analysis is important for institutions to adjust their registration process, ensuring fair access to courses for all priority groups.
Statistical Likelihood Estimation
Statistical likelihood estimation involves estimating the probability of an event occurring within a certain population. It's a fundamental concept in statistics that allows us to make predictions about future outcomes based on existing data.

Applying this to our university course registration scenario, we want to know if a student in the third priority group is likely to get more than 9 units. From the given table, we add the proportions of students in the third group that acquired 10-12 units and those who acquired more than 12. The sum is 0.09, which represents 9%. This is substantially less than 50%, which we generally consider as the threshold for 'likely'. Therefore, a student in the third priority group is statistically unlikely to receive more than 9 units in the first call. Such insights can guide students' expectations realistically and can also inform university policy for resource allocation.

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Most popular questions from this chapter

A radio station that plays classical music has a "by request" program each Saturday evening. The percentages of requests for composers on a particular night are as follows: \(\begin{array}{lr}\text { Bach } & 5 \% \\ \text { Mozart } & 21 \% \\ \text { Beethoven } & 26 \% \\ \text { Schubert } & 12 \% \\ \text { Brahms } & 9 \% \\ \text { Schumann } & 7 \% \\ \text { Dvorak } & 2 \% \\ \text { Tchaikovsky } & 14 \% \\ \text { Mendelssohn } & 3 \% \\ \text { Wagner } & 1 \%\end{array}\) Suppose that one of these requests is randomly selected. a. What is the probability that the request is for one of the three B's? b. What is the probability that the request is not for one of the two S's? c. Neither Bach nor Wagner wrote any symphonies. What is the probability that the request is for a composer who wrote at least one symphony?

Is ultrasound a reliable method for determining the gender of an unborn baby? Consider the following data on 1000 births, which are consistent with summary values that appeared in the online Journal of Statistics Education ("New Approaches to Learning Probability in the First Statistics Course" [2001]): $$\begin{array}{l|cc} \hline & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Female } \end{array} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Male } \end{array} \\ \hline \begin{array}{l} \text { Actual gender } \\ \text { is female } \end{array} & 432 & 48 \\ \begin{array}{l} \text { Actual gender } \\ \text { is male } \end{array} & 130 & 390 \\ \hline \end{array}$$ Do you think ultrasound is equally reliable for predicting gender for boys and for girls? Explain, using the information in the table to calculate estimates of any probabilities that are relevant to your conclusion.

USA Today (March 15,2001 ) introduced a measure of racial and ethnic diversity called the Diversity Index. The Diversity Index is supposed to approximate the probability that two randomly selected individuals are racially or ethnically different. The equation used to compute the Diversity Index after the 1990 census was $$\begin{aligned} 1-&\left[P(W)^{2}+P(B)^{2}+P(A I)^{2}+P(A P I)^{2}\right] \\ & \cdot\left[P(H)^{2}+P(\text { not } H)^{2}\right] \end{aligned}$$ where \(W\) is the event that a randomly selected individual is white, \(B\) is the event that a randomly selected individual is black, \(A I\) is the event that a randomly selected individual is American Indian, \(A P I\) is the event that a randomly selected individual is Asian or Pacific Islander, and \(H\) is the event that a randomly selected individual is Hispanic. The explanation of this index stated that 1\. \(\left[P(W)^{2}+P(B)^{2}+P(A I)^{2}+P(A P I)^{2}\right]\) is the probability that two randomly selected individuals are the same race 2\. \(\left[P(H)^{2}+P(\text { not } H)^{2}\right]\) is the probability that two randomly selected individuals are either both Hispanic or both not Hispanic 3\. The calculation of the Diversity Index treats Hispanic ethnicity as if it were independent of race a. What additional assumption about race must be made to justify use of the addition rule in the computation of \(\left[P(W)^{2}+P(B)^{2}+P(A I)^{2}+P(A P I)^{2}\right]\) as the probability that two randomly selected individuals are of the same race? b. Three different probability rules are used in the calculation of the Diversity Index: the Complement Rule, the Addition Rule, and the Multiplication Rule. Describe the way in which each is used.

Three friends \((\mathrm{A}, \mathrm{B}\), and \(\mathrm{C})\) will participate in a round-robin tournament in which each one plays both of the others. Suppose that \(P(\mathrm{~A}\) beats \(\mathrm{B})=.7, P(\mathrm{~A}\) beats \(\mathrm{C})\) \(=.8\), and \(P(\) B beats \(C\) ) \(=.6\) and that the outcomes of the three matches are independent of one another. a. What is the probability that \(\mathrm{A}\) wins both her matches and that B beats \(C\) ? b. What is the probability that A wins both her matches? c. What is the probability that A loses both her matches? d. What is the probability that each person wins one match? (Hint: There are two different ways for this to happen. Calculate the probability of each separately, and then add. )

Researchers at UCLA were interested in whether working mothers were more likely to suffer workplace injuries than women without children. They studied 1400 working women, and a summary of their findings was reported in the San Luis Obispo Telegram-Tribune (February 28,1995 ). The information in the following table is consistent with summary values reported in the article: $$\begin{array}{l|cccc} & & & \text { Children, } & \\ & \text { No } & \text { Children } & \text { but None } & \\ & \text { Children } & \text { Under 6 } & \text { Under 6 } & \text { Total } \\\ \hline \text { Injured on } & & & & \\ \text { the Job in } & & & & \\ 1989 & 32 & 68 & 56 & \mathbf{1 5 6} \\ \text { Not Injured } & & & & \\ \text { on the Job } & & & & \\ \text { in 1989 } & 368 & 232 & 644 & \mathbf{1 2 4 4} \\ \text { Total } & \mathbf{4 0 0} & \mathbf{3 0 0} & \mathbf{7 0 0} & \mathbf{1 4 0 0} \end{array}$$ The researchers drew the following conclusion: Women with children younger than age 6 are much more likely to be injured on the job than childless women or mothers with older children. Provide a justification for the researchers' conclusion. Use the information in the table to calculate estimates of any probabilities that are relevant to your justification.
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