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Chapter 9: Problem 31

Women in math (5.3) Of the 16,701 degrees in mathematics given by U.S. colleges and universities in a recent year, 73% were bachelor鈥檚 degrees, 21% were master鈥檚 degrees, and the rest were doctorates. Moreover, women earned 48% of the bachelor鈥檚 degrees, 42% of the master鈥檚 degrees, and 29% of the doctorates. (a) How many of the mathematics degrees given in this year were earned by women? Justify your answer. (b) Are the events 鈥渄egree earned by a woman鈥� and 鈥渄egree was a master鈥檚 degree鈥� independent? Justify your answer using appropriate probabilities. (c) If you choose 2 of the 16,701 mathematics degrees at random, what is the probability that at least 1 of the 2 degrees was earned by a woman? Show your work.

Short Answer

Expert verified
(a) 7,614 (b) No, they are not independent. (c) 0.704

Step by step solution

01

Calculate Number of Degrees

First, calculate the number of each type of degree. There are 16,701 total degrees:- Bachelor鈥檚 degrees: \(0.73 \times 16,701 = 12,192.73\). Rounding gives 12,193 degrees.- Master鈥檚 degrees: \(0.21 \times 16,701 = 3,507.21\). Rounding gives 3,507 degrees.- Doctorate degrees are the remaining, which are \(16,701 - 12,193 - 3,507 = 1,001\).
02

Calculating Women's Degrees

Now, calculate the number of degrees earned by women:- Women with Bachelor鈥檚: \(0.48 \times 12,193 = 5,851.44\). Rounding gives 5,851.- Women with Master鈥檚: \(0.42 \times 3,507 = 1,472.94\). Rounding gives 1,473.- Women with Doctorates: \(0.29 \times 1,001 = 290.29\). Rounding gives 290.Adding these, the total number of degrees earned by women is \(5,851 + 1,473 + 290 = 7,614\).
03

Calculate Probabilities for Independence

Calculate the necessary probabilities:- Probability a degree is earned by a woman, \(P(W) = \frac{7,614}{16,701} \approx 0.456\).- Probability a degree is a Master's, \(P(M) = \frac{3,507}{16,701} \approx 0.21\).- Probability a degree is a Master鈥檚 degree and earned by a woman, \(P(W \cap M) = \frac{1,473}{16,701} \approx 0.088\).To determine independence, check if \(P(W \cap M) = P(W) \cdot P(M)\). Calculating \(P(W) \cdot P(M) = 0.456 \times 0.21 \approx 0.0958\). Since \(0.088\) does not equal \(0.0958\), the events are not independent.
04

Probability of at Least One Woman's Degree

Use the complement rule. First, calculate the probability that both selected are from men:- Degrees not earned by women are \(16,701 - 7,614 = 9,087\).- Probability first degree is not by a woman: \(\frac{9,087}{16,701}\).- Probability second degree is also not by a woman: \(\frac{9,086}{16,700}\).Therefore, the probability both are not by women is \(\frac{9,087}{16,701} \times \frac{9,086}{16,700} \approx 0.296\).Thus, the probability at least one is by a woman is \(1 - 0.296 = 0.704\).

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Key Concepts

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

Statistics
Statistics is all about collecting, analyzing, interpreting, and presenting data. In this particular exercise, we are looking at the distribution of degrees in mathematics among different educational levels鈥攂achelor鈥檚, master鈥檚, and doctorates鈥攁nd the percentage of those degrees earned by women. This kind of statistical analysis helps us understand trends in education and gender distribution in mathematical fields.

By calculating the percentages and using them to find actual numbers, we extract meaningful data from a broad population. These statistics can further be visualized or formed into reports, showing how more or fewer women are entering and advancing in mathematics.

Furthermore, statistics helps in decision making by providing evidence-based insights. In educational contexts, such quantitative analysis can guide policy improvements for gender equality and create more opportunities in STEM fields for all genders.
Independence of events
Two events are considered independent if the occurrence of one does not affect the probability of the other. In this scenario, we're checking the independence between receiving a master鈥檚 degree and the degree being earned by a woman.

To test independence, we compare the probability of the intersection of two events, in this case, master's degree and women's degree, with the product of the individual probabilities of each event. If the product equals the intersection probability, the events are independent.

Here, we found that the probability of a master鈥檚 degree being earned by a woman, when multiplied with the general percentage of women鈥檚 degrees, is not equal to the intersection probability. This tells us that these events are not independent. Understanding such dependencies is crucial in probability and statistics as it influences how we approach data analysis and interpretation.
Mathematical degrees
Mathematical degrees are awarded at different levels of education鈥攂achelor鈥檚, master鈥檚, and doctoral. Each level comes with its own requirements and scope of study. In the problem, we see their distribution in percentage form: bachelor's degrees being the majority, followed by master鈥檚, and then doctorates.

This distribution is typical in higher education where bachelor's degrees naturally have a higher count due to being the foundational level, while advanced degrees like master's and doctorates require more specialization.
  • Bachelor鈥檚 degree: Usually the first level of higher education, provided as an entry point into the field.
  • Master鈥檚 degree: An advanced degree for specialized knowledge after completing a bachelor鈥檚.
  • Doctorate: The highest level of academic achievement, involving original research.
The exercise also examines gender distribution within these degrees, showcasing the reality of women鈥檚 representation in mathematics education.
Complement rule in probability
The complement rule in probability is a helpful tool when dealing with probabilities of events. It states that the probability of an event not occurring plus the probability of it occurring equals one.

For example, in part (c) of the exercise, we are asked to find the probability that at least one degree of the two randomly chosen was earned by a woman. Instead of calculating the direct probability, we compute the complement鈥� the probability that neither degree was earned by a woman, then subtract it from one.

This method simplifies complex probability problems and can be crucial when dealing with large datasets like the one in this problem.
  • The complement simplifies by reducing direct calculations in scenarios with multiple outcomes.
  • Direct calculation of probabilities sometimes involves complex scenarios, whereas the complement simplifies it to manageable terms.
In educational settings, understanding the complement rule can greatly enhance problem-solving skills and logical reasoning.

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Most popular questions from this chapter

Do you have ESP? A researcher looking for evidence of extrasensory perception (ESP) tests 500 subjects. Four of these subjects do significantly better (P 0.01) than random guessing. (a) Is it proper to conclude that these four people have ESP? Explain your answer. (b) What should the researcher now do to test whether any of these four subjects have ESP?

Losing weight A Gallup Poll found that 59% of the people in its sample said 鈥淵es鈥� when asked, 鈥淲ould you like to lose weight?鈥� Gallup announced: 鈥淔or results based on the total sample of national adults, one can say with 95% confidence that the margin of (sampling) error is \(\pm 3\) percentage points." \(^{\prime \prime 6}\) Can we use this interval to conclude that the actual proportion of U.S. adults who would say they want to lose weight differs from 0.55\(?\) Justify your answer.

Is this what P means? When asked to explain the meaning of the P-value in Exercise 13, a student says, 鈥淭his means there is only probability 0.01 that the null hypothesis is true.鈥� Explain clearly why the student鈥檚 explanation is wrong.

Power A drug manufacturer claims that fewer than 10% of patients who take its new drug for treating Alzheimer鈥檚 disease will experience nausea. To test this claim, a significance test is carried out of $$ \begin{array}{l}{H_{0} : p=0.10} \\ {H_{a} : p<0.10}\end{array} $$ You learn that the power of this test at the 5\(\%\) significance level against the alternative \(p=0.08\) is 0.64 (a) Explain in simple language what "power \(=0.64^{\prime \prime}\) means in this setting. (b) You could get higher power against the same alternative with the same \(\alpha\) by changing the number of measurements you make. Should you make more measurements or fewer to increase power? Explain. (c) If you decide to use \(\alpha=0.01\) in place of \(\alpha=\) \(0.05,\) with no other changes in the test, will the power increase or decrease? Justify your answer. (d) If you shift your interest to the alternative \(p=\) 0.07 with no other changes, will the power increase or decrease? Justify your answer.

Study more! The significance test in Exercise 76 yields a P-value of 0.0622. (a) Describe a Type I and a Type II error in this setting. Which type of error could you have made in Exercise 76? Why? (b) Which of the following changes would give the test a higher power to detect \(\mu=120\) minutes: using \(\alpha=0.01\) or \(\alpha=0.10 ?\) Explain.
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