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Toronto drivers have been going to small towns in Ontario (Canada) to take the drivers' road test, rather than taking the test in Toronto, because the pass rate in the small towns is \(90 \%\), which is much higher than the pass rate in Toronto. Suppose that every day, 100 people independently take the test in one of these small towns. 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 the test will be as low as and as high as d. If you found that on one day, 89 out of 100 passed the test, would you consider this to be a very high number?

Step by step solution

01

Calculate the Expected Value

The expected value (E) or mean is the sum of all possible outcomes each multiplied by its probability of occurrence. In this case, the expected value can be calculated as follows: \(E = np = 100*0.9 = 90\).

02

Calculate the Standard Deviation

The standard deviation measures the amount of variation or dispersion in a set of values. Under binomial distribution, standard deviation can be calculated using the formula \(\sqrt{np(1 - p)}\). So, \(SD = \sqrt{100*0.9*0.1} = 3\).

03

Apply Empirical Rule for 95%

According to the Empirical Rule, 95% of the data lies within 2 standard deviations of the mean. In this case, about 95% of days the number of people passing the test will be as low as \(90 - 2*3 = 84\) and as high as \(90 + 2*3 = 96\).

04

Analyze the Given Day

Given that on one day, 89 out of 100 passed the test. This value lies within 1 standard deviation of the mean (between 87 and 93), so it would not be considered as a very high number.

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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 is a key concept in probability and statistics. It represents the average or mean outcome if an experiment is repeated many times. In a binomial distribution, the expected value can be calculated using the formula:

• \( E = np \)

where \( n \) is the number of trials and \( p \) is the probability of success in each trial.
The expected value tells us the central tendency of the data. For the driver's test example, with 100 participants per day and a 90% pass rate, the expected number of people passing is:

• \( E = 100 \times 0.9 = 90 \).

This means that on average, 90 people are expected to pass the test daily.

Standard Deviation

Standard deviation measures how much individuals in a dataset differ from the mean value. In the context of a binomial distribution, it shows the variability or spread of the number of successes in the trials.
The formula for standard deviation in a binomial distribution is:

• \( SD = \sqrt{np(1-p)} \)

Here, \( n \) is the number of trials, \( p \) is the probability of success, and \( (1-p) \) is the probability of failure.
For the driving test case, with an expected value of 90 passers:

• \( SD = \sqrt{100 \times 0.9 \times 0.1} = 3 \).

So, on a typical day, the number of successes is expected to vary by about 3 students.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule stating that for a normal distribution:

• 68% of data falls within one standard deviation of the mean.
• 95% falls within two standard deviations.
• 99.7% falls within three standard deviations.

For the driving test example, the rule helps define the range of passers we can expect on 95% of days. With a mean of 90 passes and a standard deviation of 3, two standard deviations mean:

• Lower Limit: \( 90 - 2 \times 3 = 84 \)
• Upper Limit: \( 90 + 2 \times 3 = 96 \)

This suggests that most days, between 84 and 96 people will pass.

Probability

Probability quantifies the likelihood of an event occurring. In this problem, we're interested in the probability of whether 89 people passing the test on a given day is unusual.
Given a mean of 90 passers and a standard deviation of 3, 89 falls within one standard deviation of the mean. Since the Empirical Rule states that 68% of values fall within this range, 89 is not particularly unusual.

• Range within 1 SD: \( 90 \pm 3 \) gives us 87 to 93.

This means there is a high probability that any number within this range, including 89, will occur frequently. Therefore, 89 passers on a day is not considered out of the ordinary.