Understandable Statistics: Concepts and Methods
Understandable Statistics: Concepts and Methods
12th Edition
ISBN: 9781337119917
Author: Charles Henry Brase, Corrinne Pellillo Brase
Publisher: Cengage Learning
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Chapter 6.5, Problem 15P

(a)

To determine

Find the probability that on a single test x<40.

(a)

Expert Solution
Check Mark

Answer to Problem 15P

The probability that a single test, x<40 is 0.0359.

Explanation of Solution

Calculation:

Z score:

The number of standard deviations the original measurement x is from the value of mean μ is measured using the z-score or z value. The formula for z score is,

z=xμσ

In the formula, x is the raw score, μ is the mean and σ is the standard deviation.

The variable x denotes the level of glucose in the blood after a 12-hour fast.

Substitute x as 40, μ as 85 and σ as 25 in the formula of z score.

z=408525=4525=1.8

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than –1.8.

  • Locate the value –1.8 in column z.
  • Locate the value 0.00 in top row.
  • The intersecting value of row and column is 0.0359.

The probability is,

P(x<1.8)=P(z<1.8)=0.0359

Hence, the probability that a single test, x<40 is 0.0359.

(b)

To determine

Comment on the probability distribution of x¯.

Find the probability that x¯<40 when two tests taken about a week apart.

(b)

Expert Solution
Check Mark

Answer to Problem 15P

The probability of x¯<40 when two tests taken about a week apart is 0.0054.

Explanation of Solution

Calculation:

Sampling distribution:

For x distribution the sampling distribution of x¯ is,

μx¯=μσx¯=σn

In the formula μ denotes the population mean, σ denotes the population standard deviation, and n denotes the sample size.

The x¯ can be converted into standard normal distribution as,

z=x¯μx¯σx¯

In the formula, x¯ is the mean raw score, μx¯ is the mean of the sampling distribution and σx¯ is the standard deviation of the sampling distribution.

When the x distribution follows normal distribution the sampling distribution of x¯ is normal regard less of the sample size considered. Hence, the x¯ distribution of sample size 2 is normally distributed.

The mean of the x distribution is μ=85. The sampling distribution has the mean same as the population x distribution. That is,

μx¯=μ=85

The value of μx¯ is 85.

The standard deviation of the distribution is,

σx¯=252=251.4142=17.68

The value of σx¯ is 17.68.

Substitute x¯ as 40, μx¯ as 85 and σx¯ as 17.68 in the formula of z score.

z=408517.68=4517.68=2.55

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than –2.55.

  • Locate the value –2.5 in column z.
  • Locate the value 0.05 in top row.
  • The intersecting value of row and column is 0.0054.

The probability is,

P(x¯<40)=P(z<2.55)=0.0054

Hence, the probability of x¯<40 when two tests taken about a week apart is 0.0054.

(c)

To determine

Comment on the probability distribution of x¯.

Find the probability of x¯<40 when three tests taken about a week apart.

(c)

Expert Solution
Check Mark

Answer to Problem 15P

The probability of x¯<40 when three tests taken about a week is 0.0009.

Explanation of Solution

Calculation:

When the x distribution follows normal distribution the sampling distribution of x¯ is normal regard less of the sample size considered. Hence, the x¯ distribution of sample size 3 is normally distributed.

The mean of the x distribution is μ=85. The sampling distribution has the mean same as the population x distribution. That is,

μx¯=μ=85

The value of μx¯ is 85.

The standard deviation of the distribution is,

σx¯=253=251.7321=14.43

The value of σx¯ is 14.43.

Substitute x¯ as 40, μx¯ as 85 and σx¯ as 14.43 in the formula of z score.

z=408514.43=4514.43=3.12

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than –3.12.

  • Locate the value –2.5 in column z.
  • Locate the value 0.05 in top row.
  • The intersecting value of row and column is 0.0009.

The probability is,

P(x¯<40)=P(z<3.12)=0.0009

Hence, the probability of x¯<40 when three tests taken about a week is 0.0009.

(d)

To determine

Comment on the probability distribution of x¯.

Find the probability of x¯<40 when five tests taken about a week apart.

(d)

Expert Solution
Check Mark

Answer to Problem 15P

The probability of x¯<40 when five tests taken about a week is 0.0009.

Explanation of Solution

Calculation:

When the x distribution follows normal distribution the sampling distribution of x¯ is normal regard less of the sample size considered. Hence, the x¯ distribution of sample size 5 is normally distributed.

The mean of the x distribution is μ=85. The sampling distribution has the mean same as the population x distribution. That is,

μx¯=μ=85

The value of μx¯ is 85.

The standard deviation of the distribution is,

σx¯=255=252.2361=11.18

The value of σx¯ is 11.18.

Substitute x¯ as 40, μx¯ as 85 and σx¯ as 11.18 in the formula of z score.

z=408511.18=4511.18=4.03

Use the Appendix II: Tables, Table 5: Areas of a Standard Normal Distribution: to obtain probability less than –4.03.

  • For values of z less than –3.49 the probability is approximately 0.000.

The probability is,

P(x¯<40)=P(z<4.03)=0.0000

Hence, the probability of x¯<40 when five tests taken about a week is approximately 0.0000.

(e)

To determine

Identify whether the probabilities decreased as n increased or not.

Explain whether the test result of x¯<40 based on five tests is extremely rare event or more likely event.

(e)

Expert Solution
Check Mark

Explanation of Solution

From part (a), the probability that a single test, x<40 is 0.0359.

From part (b), the probability of x¯<40 for n=2 tests taken about a week apart is 0.0054.

From part (c), the probability of x¯<40 for n=3 tests taken about a week apart is 0.0009.

From part (d), the probability of x¯<40 for n=5 tests taken about a week apart is 0.0000.

It can be observed that, as the number of tests taken about a week apart is increasing the probability of x¯<40 is decreasing.

The variable x denotes the level of glucose in the blood after a 12-hour fast. If the value of x is less than 40 then there would be severe excess insulin. If 5 tests were taken in a week then the probability value for x¯<40 is 0.000. That is, about 0% of times x would be less than 40 if 5 tests were taken indicating that it is certain to have excess insulin.

Hence, the test result of x¯<40 based on five tests is more likely event.

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Chapter 6 Solutions

Understandable Statistics: Concepts and Methods

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