CHAPTER 9 REVIEW



CHAPTER 9 REVIEW

1) The average adult has completed an average of 11.25 years of education with a standard deviation of 1.75 years. A random sample of 90 adults in obtained. What is the probability that the sample will have a mean

(a) greater than 11.5 years?

(b) between 11 and 11.5 years?

ANS:

The sampling distribution of x-bar has [pic]

Because the sample size is large (n=90), the central limit theorem tells us that large sample techniques are appropriate. Accordingly,

(a) P([pic]or normalcdf(11.5, 1E99, 11.25, 0.184)

(b) normalcdf(11, 11.5, 11.25, 0.184) = 0.8258

2) Over the years, the scores on the final exam for AP Calculus have been normally distributed with a mean of 82 and a standard deviation of 6. The instructor thought that this year’s class was quite dull and, in fact, they only averaged 79 on their final. Assuming that this class is a random sample of 32 students from AP Calculus, what is the probability that the average score on the final for this class is no more than 79? Do you think the instructor was right?

ANS:

P([pic]or normalcdf(-1E99, 79, 82, 1.06)

If this group really were typical, there is less than a 1% chance of getting an average this low by random chance alone. That seems unlikely, so we have some pretty good evidence that the instructor was correct.

3) Harold fails to study for his statistical final. The final has 100 multiple choice questions, each with 5 choices. Harold has no choice but to guess randomly at all 100 questions. What is the probability that Harold will get at least 30% on the test?

ANS:

Because 100(0.2) and 100(0.8) are both greater than 10, we can use the normal approximation to the sampling distribution of p-hat. Because p = 0.2, the sampling distribution of p-hat has μ = 0.2 and[pic]

Therefore, P([pic]or normalcdf(0.3, 1E99, 0.2, 0.04)

Harold should have studied.

4) What is the probability that a sample of size 35 drawn from a population with mean 65 and standard deviation 6 will have a mean less than 64?

ANS:

The sample size is large enough that we can use large-sample procedures. Hence,

[pic]or normalcdf(-1E99, 64, 65, 1.014)

5) Approximately 10% of the population of the United States is known to have blood type B. If this is correct, what is the probability that between 11% and 15% of a random sample of 50 adults will have type B blood?

ANS:

If p = 0.10, [pic]

Then, [pic]or normalcdf(0.11, 0.15, 0.10, 0.042)

6) A tire manufacturer claims that his tired will last 40,000 miles with a standard deviation of 5000 miles.

(a) Assuming that the claim is true, describe the sampling distribution of the mean lifetime of a random sample of 160 tires. “Describe” means discuss center, spread, and shape.

(b) What is the probability that the mean lifetime of the sample of 160 tires will be less than 39,000 miles? Interpret the probability in terms of the truth of the manufacturer’s claim.

ANS:

[pic]miles

(a) With n = 160, the sampling distribution of x-bar will be approximately normally distributed with mean equal to 40,000 miles and standard deviation 395.28 miles.

(b) [pic]or normalcdf(-1E99, 39000, 40000, 395.28)

If the manufacturer is correct, there is only about a 0.6% chance of getting an average this low or lower. That makes is unlikely to be just a chance occurrence and we should have some doubts about the manufacturer’s claim.

7) Crabs off the coast of Northern California have a mean weight of 2 lbs. with a standard deviation of 5 oz. A large trap captures 35 crabs.

(a) Describe the sampling distribution for the average weight of a random sample of 35 crabs taken from this population.

(b) What would the mean weight of a sample of 35 crabs have to be in order to be in the top 10% of all such samples?

ANS:

[pic]

(a) With samples of size 35, the central limit theorem tells us that the sampling distribution of x-bar is approximately normal with mean 32 oz. and standard deviation 0.338 oz.

(b) In order for x-bar to be in the top 10% of samples, it would have to be at the 90th percentile, which tells us that its z-score is 1.28. Hence, [pic]

Solving, we have x-bar = 32.43 oz. A crab would have to weigh at least 32.43 oz., or about 2 lb. 7 oz., to be in the top 10% of samples of this size.

8) A certain type of light bulb is advertised to have an average life of 1200 hours. If, in fact, light bulbs of this type only average 1185 hours with a standard deviation of 80 hours, what is the probability that a sample of 100 bulbs will have an average life of at least 1200 hours?

ANS:[pic]

[pic]or normalcdf(1200, 1E99, 1185, 8)

9) Opinion polls in 2002 showed that about 70% of the population had a favorable opinion of President Bush. That same year, a simple random sample of 600 adults living in the San Francisco Bay Area showed found only 65% that had a favorable opinion of President Bush. What is the probability of getting a rating of 65% or less in a random sample of this size if the true proportion in the population was 0.70?

ANS:

If p = 0.70, [pic]

Then, [pic]or normalcdf(-1E99, 0.65, 0.70, 0.019)

10) A polling organization asks an SRS of 1500 employees if they took things from their place of work for personal use. It is believed that on a national level, 35% of employees engage in white-collar crimes and take things from their place of employment. What is the probability that a random sample of 1500 will give a result within 2 percentage points of the true population proportion of 35%?

ANS:

n = 1500 and p = 0.35. We are interested to find[pic]

Can we use the normal distribution to approximate the sampling distribution of p-hat?

np = 1500(0.35) = 525 and n(1-p) = 1500(0.65) = 975. Both are much larger than 10.

Since the conditions are met, the normal approximation is an appropriate model.

The sampling distribution of p-hat has [pic]

[pic]or normalcdf(0.33, 0.37, 0.35, 0.0123)

11) In a large high school of 2500 students, 21% of them are seniors. A simple random sample of 150 students is taken and the proportion of seniors calculated. What are the mean and standard deviation of the sample proportion, p-hat?

ANS:

In this case, the population proportion is p = 0.21 and the sample size is n = 150. The mean of the sample proportion p-hat is [pic]and the standard deviation is [pic]

12) In a large high school of 2500 students, 21% of them are seniors. A simple random sample of 150 students is taken and the proportion of seniors calculated. What is the probability that the sample will contain less than 15% seniors?

ANS:

Since the binomial conditions np = 150(0.21) = 31.5 and n(1-p) = 150(0.79) = 118.5 are both greater than 10—the sampling distribution of p-hat can be approximated by a normal distribution with mean 0.21 and standard deviation 0.033 (the results from (11)).

[pic]or normalcdf(-1E99, 0.15, 0.21, 0.033)

Thus, the chance of obtaining a sample with less than 15% seniors is about 3.6%--not very likely.

13) In a large high school of 2500 students, the mean number of cars owned by students’ families is 2.35 with a standard deviation of 1.06. A simple random sample of 36 students is taken and the mean number of cars owned is calculated.

(a) What are the mean and standard deviation of the sample mean, x-bar?

(b) What is the probability that the sample mean is greater than 2.5 cars?

(c) Below what value is the lowest 5% of all possible sample means?

ANS:

(a) The population mean is μ = 2.35 cars, the standard deviation is σ = 1.06 cars, and the sample size is n = 36. The mean of the sample mean x-bar is [pic]

(b) [pic]or normalcdf(2.5, 1E99, 2.35, 0.177)

(c) The 5th percentile in Table A corresponds to a z-score of -1.65. Therefore, [pic]

Solving for x-bar, we get 2.958. The lowest 5% of all sample means will be below 2.958 cars.

14) The GPAs of graduating students at a large university are normally distributed, with a mean GPA of 2.8 and a standard deviation of 0.5. A random sample of 50 students is taken from all the graduating students.

(a) Find the probability that the mean GPA of the sampled students is above 3.0.

(b) Find the probability that the mean GPA of sampled students is between 2.7 and 3.0.

ANS:

Sample size 50 is sufficiently large for us to assume approximate normality for the sampling distribution of x-bar with [pic] In other words, X ~ N(2.8, 0.071) approximately.

(a) [pic]or normalcdf(3.0, 1E99, 2.8, 0.071)

There is less than a 1% chance (0.24%) that the mean GPA of the 50 sampled students will exceed 3.0.

(b) [pic]or normalcdf(2.7, 3.0, 2.8, 0.071)

There is almost a 92% chance that the mean GPA of the 50 sampled students will be between 2.7 and 3.0.

15) The mathematics department at a state university notes that the SAT math scores of high school seniors applying for admission into their program are normally distributed with a mean of 610 and standard deviation of 50.

(a) What is the probability that a randomly chosen applicant to the department has an SAT math score above 700?

(b) What is the shape, mean, and standard deviation of the sampling distribution of the mean of a sample of 40 randomly selected applicants?

(c) What is the probability that the mean SAT math score in an SRS of 40 applicants is above 625?

(d) Would your answers to (a), (b), or (c) be affected if the original population of SAT math scores were highly skewed instead of normal? Explain.

ANS:

(a) [pic]or normalcdf(700, 1E99, 610, 50)

(b) It is roughly normal with N(610, [pic]=N(610, 7.91)

(c) [pic]or normalcdf(625, 1E99, 610, 7.91)

(d) The answer to part (a) would be affected because it assumes a normal population. The other answers would not be affected because for large enough n, the central limit theorem gives that the sampling distribution will be roughly normal regardless of the distribution of the original population.

16) A jar of Jiffy Peanut Butter (labeled as a 32 oz. jar) is selected randomly off the end of the assembly line. The weight of the jar is measured. If X measures this weight then X has the N(32.3, 0.40) distribution (the filling machine is calibrated this way). That is, the mean fill is 32.3 oz., the standard deviation of the fills is 0.40 oz.

(a) Identify the response variable being measured. Is it categorical or quantitative?

(b) Identify the population and parameters.

(c) What is the average fill of all Jiffy jars? Is this value a statistic?

(d) What’s the probability a randomly selected jar is filled with less than 32.0 fl oz? Such a can would be “under volume.”

(e) What proportion of all Jiffy jars are under-volume?

(f) Consider a simple random sample of 6 jars. Let x-bar be the sample mean weight for these 6 jars. Is x-bar a parameter or a statistic?

(g) What are the mean and standard deviation of the (sampling) distribution for the sample mean?

(h) Find the probability the sample mean fill of the 6 jars is less than 32.0 fl oz.

(i) Now take a simple random sample of 24 jars. What are the mean and standard deviation of the distribution for x-bar?

(j) Find the probability the sample mean fill of the 24 jars is less than 32.0 fl oz.

ANS:

(a) Response variable = weight of the jar (quantitative.)

(b) Population = measurement of jars of Jiffy Peanut Butter.

Parameter = 32 oz.

(c) Average = 32.3 oz.

No, it’s not a statistic.

(d) P(X ................
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