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Unformatted text preview: Student: Grady Silnonton Course: Math119: Elementary Statistics  Spring 2.010  CRN: 49239
Instructor: Shawn Parvini  16 weeks Date: 3/18/10 Book: Triola: Elementary Statistics, 11e
Time: 4:13 PM The capacity of a lift is 16 people or 2560 pounds. The capacity will be exceeded if 16 people have weights with a mean
greater than 2560! 16 = 160 pounds. Suppose the people have weights that are normally distributed with a mean of
165 lb and a standard deviation of 31 lb. a. Find the probability that if a person is randomly selected, his weight will be greater than 160 pounds. Notice that an individual value from a normally distributed population has been chosen. Therefore, use the population
distribution to determine the probability. xu First, convert the height to the corresponding 2 score using 2 = and the population distribution statistics. 160— 165
31
=5 —0.16 You can either use a standard normal distribution table or technology to ﬁnd the area under the normal curve. For this
explanation, a standard normal distribution table is used. The probability is the area to the right of z = — 0.16 under the standard normal
distribution. First ﬁnd the area to the left from a table. The area to the left is approximately 0.4364. §=160
n;=165 Subtract this value from 1 to ﬁnd the area to the right. 1  0.4364 = 0.5636 Therefore, the probability that a randomly selected person's weight is greater than 160 lb is approximately 0.5636.
I}. Find the probability that 16 randomly selected people will have a mean that is greater than 160 pounds. In this case, the desired probability is for the mean of a sample of 16 people. Therefore, use the central limit theorem. According to the central limit theorem, the distribution of sample means i will have a mean given by 1.1; = u and a 6
standard deviation given by 0'; = T.
n The mean of the distribution of sample means E is the same as the population mean, so u;= 165. Apply the deﬁnition for the standard deviation of the distribution of the sample means for a sample size of 16. Page 1 Student: Grady Simonton Course: Math119: Elementary Statistics  Spring 2010  CRN: 49239
Instructor: Shawn Parvini  16 weeks Date: 3/18/10 Book: Triola: Elementary Statistics, 11e
Time: 4:13 PM 0'
XV?
31 3! Therefore, the distribution of sample means i for a sample size of 16 is approximately normal with a mean 1.1; = 165 and 3 l _
a standard deviation o;= m . Use these values to compute the corresponding 2 score for x = 160. _ 160— 165 2——
31Hl16 =5  0.65 The probability is the area to the right of z = — 0.65 under the standard normal
distribution. First ﬁnd the area to the left using a table. The area to the left is approximately 0.2578. §=160
u;=165 Subtract this area ﬁ‘om l to ﬁnd the area to the right.
1  0.2578 = 0.7422 Therefore, the probability that the mean of 16 randomly selected people is greater than 160 pounds is approximately
0.2422. c. Does the lift appear to have the correct weight limit? Why or why not?
If the weight limit is correct, then the operator would not have to be concerned about the weight of the people on the lift. He would simply need to ensure that the number of people does not exceed the stated capacity. Use the probability that
the mean of 16 randomly selected people is greater than 160 to determine if this is a reasonable assumption. Page 2 ...
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