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Chapter 3: Problem 124

Use the following information to answer the next two exercises. The percent of licensed U.S. drivers (from a recent year) that are female is 48.60. Of the females, 5.03% are age 19 and under; 81.36% are age 20鈥�64; 13.61% are age 65 or over. Of the licensed U.S. male drivers, 5.04% are age 19 and under; 81.43% are age 20鈥�64; 13.53% are age 65 or over. Suppose that 10,000 U.S. licensed drivers are randomly selected. a. How many would you expect to be male? b. Using the table or tree diagram, construct a contingency table of gender versus age group. c. Using the contingency table, find the probability that out of the age 20鈥�64 group, a randomly selected driver is female.

Short Answer

Expert verified
a. 5,140 drivers are expected to be male. b. Contingency table is constructed for gender vs age group. c. The probability is approximately 0.486.

Step by step solution

01

Calculate Expected Number of Male Drivers

First, to find out how many of the 10,000 randomly selected drivers are expected to be male, we use the fact that 48.60% of drivers are female. Thus, the percentage of male drivers is 100% - 48.60% = 51.40%. Therefore, the expected number of male drivers is 51.40% of 10,000, which is calculated as:\[\text{Male Drivers} = \frac{51.40}{100} \times 10,000 = 5,140.\]
02

Calculate Expected Number of Female Drivers by Age Group

We already know 4,860 drivers will be female (since 48.60% of 10,000 are female). We need to determine how many of these are in each age group:- Age 19 and under: 5.03% of female drivers are in this group. Hence, we calculate it as:\[\text{Female Age 19 and Under} = \frac{5.03}{100} \times 4,860 = 244.458.\]- Age 20-64: 81.36% of female drivers fall into this group, calculated by\[\text{Female Age 20-64} = \frac{81.36}{100} \times 4,860 = 3,953.496.\]- Age 65 and over: 13.61% of female drivers are in this group, calculated by\[\text{Female Age 65+} = \frac{13.61}{100} \times 4,860 = 661.946.\]
03

Calculate Expected Number of Male Drivers by Age Group

Similarly, we calculate how many of the 5,140 male drivers fall into each age group:- Age 19 and under: 5.04% of male drivers, calculated by\[\text{Male Age 19 and Under} = \frac{5.04}{100} \times 5,140 = 258.744.\]- Age 20-64: 81.43% of male drivers, calculated by\[\text{Male Age 20-64} = \frac{81.43}{100} \times 5,140 = 4,184.102.\]- Age 65 and over: 13.53% of male drivers, calculated by\[\text{Male Age 65+} = \frac{13.53}{100} \times 5,140 = 694.154.\]
04

Construct the Contingency Table

Now, we create a contingency table showing the distribution of drivers by gender and age group: | Gender | Age 19 and Under | Age 20-64 | Age 65 and Over | Total | |--------|------------------|-----------|-----------------|--------| | Male | 259 | 4,184 | 694 | 5,137 | | Female | 244 | 3,953 | 662 | 4,859 | | Total | 503 | 8,137 | 1,356 | 10,000 | Note: Rounded values might slightly differ due to rounding from earlier calculations.
05

Calculate Probability of a Female in Age Group 20-64

To find the probability that a driver aged 20-64 is female, we use the formula:\[P( ext{Female | Age 20-64}) = \frac{ ext{Number of Females Age 20-64}}{ ext{Total Drivers Age 20-64}}.\]Using the table,\[P( ext{Female | Age 20-64}) = \frac{3,953}{8,137} \approx 0.486.\]

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Key Concepts

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

Probability
In statistics, probability is the basis for making predictions about future events based on known data. It helps us to express how likely it is for an event to occur. In the context of this problem, probability plays a crucial role in determining the likelihood of selecting a female driver from a specific age group.
  • Probability Formula: To find the probability of an event A, you can use the formula: \( P(A) = \text{Number of favorable outcomes} / \text{Total possible outcomes} \).

  • Application: For instance, to find the probability that a driver aged 20-64 is female, we would calculate based on how many females fall into that age group versus the total number of drivers within the same age range.
Understanding probability allows us to infer patterns and make decisions based on statistical evidence. It aids in interpreting complex data, making it easier to draw meaningful conclusions.
Age Group Analysis
Age group analysis involves breaking down data into segments based on age ranges. This allows for more detailed insights into the characteristics and trends within each segment. In this example, we are categorizing licensed drivers into age groups of 19 and under, 20-64, and 65 and over, further categorizing by gender.
  • Importance: By analyzing data across these age brackets, organizations and researchers can identify trends such as which age group holds the majority of drivers and how this might change over time.

  • Gender Differences: Studying the distinction between ages for males and females is useful in identifying demographic differences, which can drive policy or marketing decisions.
Such detailed analysis helps uncover specific characteristics of the population, transforming raw data into impactful insights.
Expected Number Calculation
Expected number calculations are fundamental in predicting outcomes given a particular probability distribution. It helps in estimating how many occurrences of a certain event can be anticipated within a set sample size.
  • Formula: The expected number of a specific category is calculated by multiplying the probability of that category by the total number of subjects. For instance, the expected number of male drivers is found by multiplying the proportion of male drivers by the total number of drivers. \[ \text{Expected Number} = \left( \frac{\text{Rate of Occurrence}}{100} \right) \times \text{Total Sample Size} \]

  • Applying to Data: In this scenario, by applying the proportion of age-specific and gender-specific distributions, we calculated how many drivers fit into each category.
This method of calculation is indispensable for making practical predictions based on statistical data, helping in fields ranging from public policy to business strategy.

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Most popular questions from this chapter

After Rob Ford, the mayor of Toronto, announced his plans to cut budget costs in late 2011, the Forum Research polled 1,046 people to measure the mayor鈥檚 popularity. Everyone polled expressed either approval or disapproval. These are the results their poll produced: 鈥� In early 2011, 60 percent of the population approved of Mayor Ford鈥檚 actions in office. 鈥� In mid-2011, 57 percent of the population approved of his actions. 鈥� In late 2011, the percentage of popular approval was measured at 42 percent. a. What is the sample size for this study? b. What proportion in the poll disapproved of Mayor Ford, according to the results from late 2011? c. How many people polled responded that they approved of Mayor Ford in late 2011? d. What is the probability that a person supported Mayor Ford, based on the data collected in mid-2011? e. What is the probability that a person supported Mayor Ford, based on the data collected in early 2011?

A special deck of cards has ten cards. Four are green, three are blue, and three are red. When a card is picked, its color of it is recorded. An experiment consists of first picking a card and then tossing a coin. a. List the sample space. b. Let A be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A). c. Let B be the event that a red or green is picked, followed by landing a head on the coin toss. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification. d. Let C be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events A and C mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.

Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange. Let B = the event of getting a blue jelly bean Let G = the event of getting a green jelly bean. Let O = the event of getting an orange jelly bean. Let P = the event of getting a purple jelly bean. Let R = the event of getting a red jelly bean. Let Y = the event of getting a yellow jelly bean. Find P(G)

What is the probability of drawing a red card in a standard deck of 52 cards?

Use the following information to answer the next two exercises. Suppose that you have eight cards. Five are green and three are yellow. The cards are well shuffled. Suppose that you randomly draw two cards, one at a time, with replacement. Let G1 = first card is green Let G2 = second card is green a. Draw a tree diagram of the situation. b. Find P(G1 AND G2). c. Find P(at least one green). d. Find P(G2|G1).
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