4.4-4.5 Notes

Sta220 - Statistics Mr. Smith Room 310 Class #13

Section 4.4-4.5

The graphical form of the probability distribution for a continuous random variable is a smooth curve.

Definition

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Figure 4.11 A probability f(x) for a continuous random variable x

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One of the most common observed continuous random variable has a bell-shaped probability distribution (or bell curve).

Figure 4.13 A normal probability distribution

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Figure 4.14 Several normal distributions with different means and standard deviations

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The normal distribution plays a very important role in the science of statistical inference. You can determine the appropriateness of the normal approximation to an existing population of data by comparing the relative frequency distribution of a large sample of the data with the normal probability distribution.

To graph the normal curve, we have to know the numerical values of 𝜇 and 𝜎. Computing the area over intervals under the normal probability distribution is difficult task, so we will use the computed areas listed in Table III of Appendix A.

Figure 4.15 Standard normal distribution: m = 0, s = 1

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Definition

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Table 4.5

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Example 4.15A Find the probability that the standard normal random variable z falls between -1.33 and +1.33.

Figure 4.16 Areas under the standard normal curve for Example 4.15

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Solution

Example 4.15B

Find the probability that the standard normal random variable z falls between 1.00 and 2.50.

Solution

Example 4.16 Find the probability that a standard normal random variable exceeds 1.64; that is, find P(z > 1.64).

Figure 4.18 Areas under the standard normal curve for Example 4.16

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Solution

Example 4.17 Find the probability that a standard normal random variable lies to the left of .67.

Figure 4.19 Areas under the standard normal curve for Example 4.17

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Solution

To apply Table III to a normal random variable x with any mean 𝜇 and any standard deviation 𝜎, we must first convent the value of x to a z-score.

This z-score transformation allows all normal distributions to be solved with the use of the standard normal distribution.

Definition

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Example 18 Suppose the random variable x is best described by a normal distribution with 𝜇 = 25 and 𝜎 = 5. Find the z-score that corresponds to the x value: x = 37.5.

Solution

Example 19 Find a value 𝑧0 of the standard normal random variable z such that:

a) 𝑃 𝑧 ≤ 𝑧0 = 0.0401 b) 𝑃(−𝑧0 ≤ 𝑧 ≤ 𝑧0 ) = .90

Solution

Example 4.20 Suppose x is a normal distributed random variable with 𝜇 = 30 and 𝜎 = 8. Find a value 𝑥0 of the random variable x such that:

a) 𝑃(𝑥 ≥ 𝑥0 ) = .5. b) 80% of the values x are less than 𝑥0 .

Solution

STOP

Sta220 - Statistics Mr. Smith Room 310 Class #14

Section 4.4-4.5(Part 2)

Problem 4. 21 Assume that the length of time, x, between charges of a cellular phone is normally distributed with a mean of 10 hours and a standard deviation of 1.5 hours. Find the probability that the cell phone will last between 8 and 12 between charges.

Solution

Procedure

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Problem 4. 22 Suppose an automobile manufacturer introduces a new model that has an advertised mean in-city mileage of 27 miles per gallon. Although such advertisements seldom report any measure of variability, suppose you write the manufacturer for details on the tests and you find that the standard deviation is 3 miles per gallon. This information leads you to formulate a probability model for the random variable x, the in-city mileage for this car model. You believe that the probability distribution of x can be approximated by a normal distribution with a mean of 27 and a standard deviation of 3.

a. If you were to buy this model of automobile, what is the probability that you would purchase one that averages less than 20 miles per gallon for in-city driving? b. Suppose you purchase one of these new models and it does get less than 20 miles per gallon for in-city driving. Should you conclude that you probability model is incorrect?

Solution Part a.

Figure 4.21 Area under the normal curve for Example 4.21

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Solution Part B. There are two possibilities that could exists: 1. The probability model is correct. You simply were unfortunate to have purchased one of the cars in the 1% that get less than 20 miles per gallon in the city. 2. The probability model is incorrect. Perhaps the assumption of a normal distribution is unwarranted, or the mean of 27 is an overestimate, or the stand deviation of 3 is an underestimate, or some combination of these errors occurred.

Keep in mind we have no way of knowing with certainty which possibility is correct, but the evidence points to the second on. We are again relying on the rare-event approach to statistical inference that we introduced earlier. In applying the rare-event approach, the calculated probability must be small (say, less than or equal to 0.05) in order to infer that the observed event is, indeed, unlikely.

Problem 4.22 Suppose that the scores x on a college entrance examination are normally distributed with a mean of 550 and the standard deviation of 100. A certain prestigious university will consider for admission only those applicants whose scores exceed the 90th percentile of the distribution. Find the minimum score an applicant must achieve in order to receive consideration for admission to the university.

Figure 4.22 Area under the normal curve for Example 4.22

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Solution Thus, the 90th percentile of the test score distribution is 678. That is to say, an applicant must score at least 678 on the entrance exam to receive consideration for admission by the university.