7. RESULTADOS Y DISCUSIÓN
7.2 Unión Soldada – Normalizado
Solution
Let pbe the proportion of all eligible voters who favor Maureen Webster.
Then,
The mean of the sampling distribution of the sample proportion is
The population of all voters is large (because the city is large) and the sample size is
small compared to the population. Consequently, we can assume that nN
.05. Hence,
the standard deviation of is calculated as
From the central limit theorem, the shape of the sampling distribution of is approxi-
mately normal. The probability that is less than .49 is given by the area under the nor-
mal distribution curve for
to the left of
as shown in Figure 7.18. The zvalue
for is
z
pˆ
p
s
p^.49.53
.02495496
1.60
pˆ.49
pˆ
.49,
pˆ
pˆ
pˆ
s
p^B
pq
n
B
1.5321.472
400
.02495496
pˆ
m
p^p.53
pˆ
p.53
and
q1p1.53.47
Calculating the probability that is lessthan a certain value. pˆ 0 – 1.60 z Shaded area is .0548 .49 µ = .53p^ p^ Figure 7.18 P(pˆ .49)
Thus, the required probability is
Hence, the probability that less than 49% of the voters in a random sample of 400 will
favor Maureen Webster is .0548.
.5.4452.0548
.5P11.60
6
z
6
02
EXERCISES
CONCEPTS AND PROCEDURES
7.83 If all possible samples of the same (large) size are selected from a population, what percentage of all sample proportions will be within 2.0 standard deviations of the population proportion? 7.84 If all possible samples of the same (large) size are selected from a population, what percentage of all sample proportions will be within 3.0 standard deviations of the population proportion? 7.85 For a population,N30,000 and p .59. Find the zvalue for each of the following for n100.
a. b. c. d.
7.86 For a population,N18,000 and p .25. Find the zvalue for each of the following for n70. a. b. c. d.
APPLICATIONS
7.87 The National Institute of Child Health and Human Development conducted a study of bullying in U.S. schools in which over 15,000 students in sixth through tenth grades were surveyed. Of students who have been bullied, 61.6% said that they were victimized because of their looks or speech (U.S. News & World Report, May 7, 2001). Assume that this percentage is true for all current sixth-through tenth-grade students who have been bullied. Let be the proportion in a random sample of 200 such students who will say that they are victimized because of their looks or speech. Find the probability that the value of is
a. between .60 and .66 b. greater than .64
7.88 A survey of all medium- and large-sized corporations showed that 64% of them offer retirement plans to their employees. Let be the proportion in a random sample of 50 such corporations that offer retirement plans to their employees. Find the probability that the value of will be
a. between .54 and .61 b. greater than .71
7.89 In a survey of workers by International Communications Research for J. Howard & Associates, 39% said that merit was the key to promotion, and others indicated seniority, connections, or luck (USA TODAY, October 2, 2002). Assume that 39% of all current workers feel that merit is the key to promotion, and that is the proportion of workers in a random sample of 300 who hold this view. Find the prob- ability that the value of is
a. between .35 and .45 b. greater than .36
7.90 Dartmouth Distribution Warehouse makes deliveries of a large number of products to its customers. It is known that 85% of all the orders it receives from its customers are delivered on time. Let be the proportion of orders in a random sample of 100 that are delivered on time. Find the probability that the value of will be
a. between .81 and .88 b. less than .87
7.91 Brooklyn Corporation manufactures CDs. The machine that is used to make these CDs is known to produce 6% defective CDs. The quality control inspector selects a sample of 100 CDs every week and inspects them for being good or defective. If 8% or more of the CDs in the sample are defective, the process is stopped and the machine is readjusted. What is the probability that based on a sample of 100 CDs the process will be stopped to readjust the machine?
7.92 Mong Corporation makes auto batteries. The company claims that 80% of its LL70 batteries are good for 70 months or longer. Assume that this claim is true. Let be the proportion in a sample of 100 such batteries that are good for 70 months or longer.
a. What is the probability that this sample proportion is within .05 of the population proportion? b. What is the probability that this sample proportion is less than the population proportion by
.06 or more?
c. What is the probability that this sample proportion is greater than the population proportion by .07 or more? pˆ pˆ pˆ pˆ pˆ pˆ pˆ pˆ pˆ pˆ .20 pˆ .17 pˆ .32 pˆ .26 pˆ .65 pˆ .53 pˆ .68 pˆ .56
GLOSSARY
Central limit theorem The theorem from which it is inferred that for a large sample size (n30), the shape of the sampling distribution of is approximately normal. Also, by the same theorem, the shape of the sampling distribution of is approx- imately normal for a sample for which np5 and nq5.
Consistent estimator A sample statistic with a standard
deviation that decreases as the sample size increases.
Estimator The sample statistic that is used to estimate a pop- ulation parameter.
Mean of The mean of the sampling distribution of
denoted by is equal to the population proportion p.
Mean of The mean of the sampling distribution of
denoted by is equal to the population mean .
Nonsampling errors The errors that occur during the collec- tion, recording, and tabulation of data.
Population distribution The probability distribution of the population data.
Population proportionp The ratio of the number of elements in a population with a specific characteristic to the total number of elements in the population.
mx, x, x mp^, pˆ, pˆ pˆ x
Sample proportion The ratio of the number of elements in a sample with a specific characteristic to the total number of elements in that sample.
Sampling distribution of The probability distribution of all the values of calculated from all possible samples of the same size selected from a population.
Sampling distribution of The probability distribution of all the values of calculated from all possible samples of the same size selected from a population.
Sampling error The difference between the value of a sam- ple statistic calculated from a random sample and the value of the corresponding population parameter. This type of error occurs due to chance.
Standard deviation of The standard deviation of the sam- pling distribution of denoted by is equal to when nN .05.
Standard deviation of The standard deviation of the sam- pling distribution of denoted by is equal to when nN .05.
Unbiased estimator An estimator with an expected value (or mean) that is equal to the value of the corresponding population parameter. s
1n sx, x, x 1pqn sp^, pˆ, pˆ x x pˆ pˆ pˆMathematics tells us that the sample mean, is an unbiased and consistent estimator for the population mean, . This is great news because it allows us to estimate properties of a population based on those of a sample; this is the essence of statistics. But statis- tics always makes a number of assumptions about the sample from which the mean and standard deviation are calculated. Fail- ure to respect these assumptions can introduce bias in your cal- culations. In statistics, bias means a deviation of the expected value of a statistical estimator from the parameter it estimates.
Let’s say you are a quality control manager for a refrigera- tor parts company. One of the parts that you manufacture has a specification that the length of the part be 2.0 centimeters plus or minus .025 centimeters. The manufacturer expects that the parts it receives have a mean length of 2.0 centimeters and a small variation around that mean. The manufacturing process is to mold the part to something a little bit bigger than necessary— say, 2.1 centimeters—and finish the process by hand. Because the action of cutting material is irreversible, the machinists tend to miss their target by approximately .01 centimeters, so the mean
x, length of the parts is not 2.0 centimeters, but rather 2.01 cen- timeters. It is your job to catch this.
One of your quality control procedures is to select com- pleted parts randomly and test them against specification. Unfortunately, your measurement device is also subject to variation and might consistently underestimate the length of the parts. If your measurements are consistently .01 centimeters too short, your sample mean will not catch the manufacturing error in the population of parts.
The solution to the manufacturing problem is relatively straightforward: be certain to calibrate your measurement in- strument. Calibration becomes very difficult when working with people. It is known that people tend to overestimate the num- ber of times that they vote and underestimate the time it takes to complete a project. Basing statistical results on this type of data can result in distorted estimates of the properties of your population. It is very important to be careful to weed out bias in your data, because once it gets into your calculations, it is very hard to get it out.