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Chapter 6: Problem 81

In Toronto, Canada, \(55 \%\) of people pass the drivers' road test. Suppose that every day, 100 people independently take the test. a. What is the number of people who are expected to pass? b. What is the standard deviation for the number expected to pass? c. After a great many days, according to the Empirical Rule, on about \(95 \%\) of these days, the number of people passing will be as low as and as high as (Hint: Find two standard deviations below and two standard deviations above the mean.) d. If you found that on one day, 85 out of 100 passed the test, would you consider this to be a very high number?

Short Answer

Expert verified
a. 55 people are expected to pass the test.\n b. The standard deviation for the number expected to pass is 5.\n c. On about 95% of the days, the number of people passing the test will be as low as 45 and as high as 65.\n d. Yes, 85 out of 100 passing the test would be considered a very high number.

Step by step solution

01

- Calculate the expected value

The expected value, also known as the mean, for a binomial distribution is calculated as \( n \cdot p \), where \( n \) is the number of trials (100 in this case) and \( p \) is the probability of success (0.55 in this case). So, the expected value \( E(X) \) = \( 100 \cdot 0.55 = 55 \)
02

- Calculate the standard deviation

The standard deviation for a binomial distribution is calculated as \( \sqrt{n \cdot p \cdot (1-p)} \), where \( n \) is the number of trials, \( p \) is the probability of success and \( (1-p) \) is the probability of failure. Plugging in the numbers, we get the standard deviation \( SD = \sqrt{100 \cdot 0.55 \cdot (1-0.55)} = 5 \)
03

- Apply the Empirical rule

According to the empirical rule, approximately 95% of the data lies within 2 standard deviations of the mean in a normal distribution. We will apply this concept by adding and subtracting two standard deviations from the mean. So, the range of people passing will be as low as \( 55 - 2 \cdot 5 = 45 \) and as high as \( 55 + 2 \cdot 5 = 65 \) on about 95% of the days.
04

- Compare with a given number

Given that 85 out of 100 passed the test on one day, we compare this number with our range calculated above. Since 85 is well above 65, it can be considered as a very high number, as it lies above the range wherein 95% of the data lies according to the empirical rule.

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Key Concepts

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

Expected Value
The expected value in a binomial distribution helps us predict the average outcome over several trials. Think of it as the "center" of our data in terms of probability. In our exercise, we're dealing with 100 people taking a driver's test, where each has a 55% chance of passing. We calculate the expected value using the formula:
  • \( E(X) = n \cdot p \)
Here, \( n \) is the number of trials (100 people) and \( p \) is the probability of one person passing (0.55). This gives us:
  • \( E(X) = 100 \times 0.55 = 55 \)
This means that, on average, we expect 55 out of 100 people to pass every day.
Understanding expected value helps us set realistic expectations for outcomes and gauge whether actual results are typical or not.
Standard Deviation
Standard deviation tells us how spread out the results are likely to be from the expected value. It measures the amount of variation or dispersion in our set of data. In the context of our problem, it helps us understand the variability in the number of people passing the test each day. The formula for standard deviation in a binomial distribution is:
  • \( SD = \sqrt{n \cdot p \cdot (1-p)} \)
Using \( n = 100 \), \( p = 0.55 \), and \( (1-p) = 0.45 \), we calculate:
  • \( SD = \sqrt{100 \times 0.55 \times 0.45} = 5 \)
This means most counts of people passing will vary by about 5 from the expected value of 55. Knowing the standard deviation, we can better assess unusual results.
Empirical Rule
The Empirical Rule is a handy way to gauge where most data in a distribution falls. It tells us about the spread of data in a normal distribution characterized by its mean and standard deviation. For our exercise, we're looking at how many people can be expected to pass on 95% of test days. Using the rule, approximately 95% of the data lies within two standard deviations from the mean.
  • Calculate this range by subtracting and adding two standard deviations from the mean: \( 55 \pm 2 \times 5 \).
  • Thus, the range is from 45 to 65.
This range indicates that on most days, between 45 and 65 people will pass the driver's test.
This insight helps us recognize any unusual outcomes, like a high pass number of 85 out of 100.
Probability
Probability is a measure that describes the likelihood of an event occurring. In our scenario, the probability of each person passing the test is 0.55, or 55%. When comparing day-to-day outcomes, probability helps us to assess whether something like 85 people passing in one day is typical. In the context of our driver's test, the day where 85 passed is an anomaly. It lies outside the usual range obtained by the Empirical Rule. This suggests that such an event has a low probability of occurring, considering the usual spread of results. Understanding probability helps us make informed judgments about daily occurrences and their typicality.

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Most popular questions from this chapter

The Normal model \(N(150,10)\) describes the distribution of scores on the LSAT, a standardized test required by most law schools. Which of the following questions asks for a probability, and which asks for a measurement? Identify the type of problem and then answer the given question. a. A law school applicant scored at the 60 th percentile on the LSAT. What was the applicant's LSAT score? b. A law school applicant scored 164 on the LSAT. This applicant scored higher than what percentage of LSAT test takers?

The distribution of spring high temperatures in Los Angeles is approximately Normal, with a mean of 75 degrees and a standard deviation of \(2.5\) degrees. a. What is the probability that the high temperature is less than 70 degrees in Los Angeles on a day in spring? b. What percentage of Spring day in Los Angeles have high temperatures between 70 and 75 degrees? c. Suppose the hottest spring day in Los Angeles had a high temperature of 91 degrees. Would this be considered unusually high, given the mean and the standard deviation of the distribution? Why or why not?

The Normal model \(N(69,3)\) describes the distribution of male heights in the United States. Which of the following questions asks for a probability, and which asks for a measurement? Identify the type of problem and then answer the given question. See page 316 for guidance. a. To be a member of the Tall Club of Silicon Valley a man must be at least 74 inches tall. What percentage of men would qualify for membership in this club? b. Suppose the Tall Club of Silicon Valley wanted to admit the tallest \(2 \%\) of men. What minimum height requirement should the club set for its membership criteria?

For each situation, identify the sample size \(n\), the probability of a success \(p\), and the number of success \(x\). When asked for the probability, state the answer in the form \(b(n, p, x)\). There is no need to give the numerical value of the probability. Assume the conditions for a binomial experiment are satisfied. A 2017 Gallup poll found that \(53 \%\) of college students were very confident that their major will lead to a good job. a. If 20 college students are chosen at random, what's the probability that 12 of them were very confident their major would lead to a good job? b. If 20 college students are chosen at random, what's the probability that 10 of them are not confident that their major would lead to a good job? a. If 20 college students are chosen at random, what's the probability that 12 of them were very confident their major would lead to a good job? b. If 20 college students are chosen at random, what's the probability that 10 of them are not confident that their major would lead to a good job?

Use technology or a Normal table to find each of the following. Include an appropriately labeled sketch of the Normal curve for each part with the appropriate area shaded. a. Find the probability that a \(z\) -score will be \(2.12\) or greater. b. Find the probability that a \(z\) -score will be less than \(-0.74\). c. Find the probability that a \(z\) -score will between \(1.25\) and \(2.37\).
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