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Chapter 8: Q158SE (page 452)

Consider the discrete probability distribution shown here.

x

10

12

18

20

p

.2

.3

.1

.4

a. Calculate渭,蟽2 and蟽 .

b. What isP(x<15) ?

c. Calculate 渭卤2蟽 .

d. What is the probability that xis in the interval 渭卤2蟽 ?

Short Answer

Expert verified

(a) 渭=15.4,蟽2=18.44and蟽=4.294.

(b) Px<15=0.5

(c) 渭卤2蟽=(6.812,23.988)

(d) The required probability is 0.5.

Step by step solution

01

Given information

X is a random variable.

02

Identifying the type of random variable

a.

Mean=渭

渭=鈭�i=14xjPxi=10脳0.2+12脳0.3+18脳0.1+20脳0.4=2+3.6+1.8+8=15.4

Variance=蟽2

蟽2=鈭�i=14x-渭2Px=10脳15.42脳0.2+12-15.42脳0.3+18-15.42脳0.1+20-15.42脳0.4=29.16脳0.2+11.56脳0.3+6.76脳0.1+21.16脳0.4=5.832+3.468+0.676+8.464=18.44

Standard Deviation=蟽

蟽=18.44=4.294

Hence, 渭=15.4,蟽2=18.44 and 蟽=4.294 .

03

When  p(x<15)

b.

px<15=px=10+px=12=0.2+0.3=0.5

Hence, Px<15=0.5

04

Calculating the value when  μ±2σ

c.

The interval is

渭卤2蟽=15.4卤2脳4.294=15.4卤8.588=6.812,23.988

05

Calculating P (x) when μ±2σ

d.

P6.812鈮�x鈮�23.988=P0鈮�x鈮�17.176=1-Px鈮�17.176=1-Px=18+Px=20=1-0.1+0.4=1-0.5=0.5

Hence, the required probability is 0.5.

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Most popular questions from this chapter

What is the line of means?

A paired difference experiment yielded ndpairs of observations. In each case, what is the rejection region for testing H0:渭d>2?

a. nd=12,伪=.05

b.nd=24,伪=.10

c.nd=4,伪=.025

d.nd=80,伪=.01

Optimal goal target in soccer. When attempting to score a goal in soccer, where should you aim your shot? Should you aim for a goalpost (as some soccer coaches teach), the middle of the goal, or some other target? To answer these questions, Chance (Fall 2009) utilized the normal probability distribution. Suppose the accuracy x of a professional soccer player鈥檚 shots follows a normal distribution with a mean of 0 feet and a standard deviation of 3 feet. (For example, if the player hits his target,x=0; if he misses his target 2 feet to the right, x=2; and if he misses 1 foot to the left,x=-1.) Now, a regulation soccer goal is 24 feet wide. Assume that a goalkeeper will stop (save) all shots within 9 feet of where he is standing; all other shots on goal will score. Consider a goalkeeper who stands in the middle of the goal.

a. If the player aims for the right goalpost, what is the probability that he will score?

b. If the player aims for the center of the goal, what is the probability that he will score?

c. If the player aims for halfway between the right goal post and the outer limit of the goalkeeper鈥檚 reach, what is the probability that he will score?

Question: The purpose of this exercise is to compare the variability of with the variability of .

a. Suppose the first sample is selected from a population with mean and variance . Within what range should the sample mean vary about of the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations of on each side of .

b. Suppose the second sample is selected independently of the first from a second population with mean and variance . Within what range should the sample mean vary about the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations on each side .

c. Now consider the difference between the two sample means . What are the mean and standard deviation of the sampling distribution ?

d. Within what range should the difference in sample means vary about the time in repeated independent samples of measurements each from the two populations?

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Drug content assessment. Scientists at GlaxoSmithKlineMedicines Research Center used high-performance liquidchromatography (HPLC) to determine the amountof drug in a tablet produced by the company (Analytical

Chemistry, Dec. 15, 2009). Drug concentrations (measuredas a percentage) for 50 randomly selected tablets are listedin the table below and saved in the accompanying file.

a. Descriptive statistics for the drug concentrations areshown at the top of the XLSTAT printout on the nextpage. Use this information to assess whether the dataare approximately normal.

b. An XLSTAT normal probability plot follows. Use thisinformation to assess whether the data are approximatelynormal.

91.28 92.83 89.35 91.90 82.85 94.83 89.83 89.00 84.62

86.96 88.32 91.17 83.86 89.74 92.24 92.59 84.21 89.36

90.96 92.85 89.39 89.82 89.91 92.16 88.67 89.35 86.51

89.04 91.82 93.02 88.32 88.76 89.26 90.36 87.16 91.74

86.12 92.10 83.33 87.61 88.20 92.78 86.35 93.84 91.20

93.44 86.77 83.77 93.19 81.79

Descriptive statistics(Quantitative data)

Statistic

Content

Nbr.of Observation

50

Minimum

81.79

Maximum

94.83

1st Quartile

87.2725

Median

89.375

3rd Quartile

91.88

Mean

89.2906

Variance(n-1)

10.1343

Standard deviation(n-1)

3.1834
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