 # Problem 01: Normal Distribution: Test Times The length of time, x, ...

## Question

Problem 01: Normal Distribution: Test Times The length of time, x, it takes to finish a statistics progress quiz is normally distributed with a mean of 18.0 minutes and a standard deviationd of 1.3 minutes. The time limit for the quiz is set at 20 minutes. For a class of 38 students, how many students will finish the test?
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:
Number of students: show your calculation

Problem 02 Sampling Distributions: Sleep Studies The population of the amount of time that an adult sleeps is normally distributed with a mean of 6.8 hours and a standard deviation of 1.4 hours (based on data from multiple sources, including a Gallup poll and the American Journal of Epidemiology). A common recommendation is that we get between 7 and 9 hours of sleep each night.

(a) For someone randomly selected, find the probability that he/she gets between 7 and 9 hours of sleep in a night.
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:

(b) If we randomly collect a sample of 5 adults, what is the probability that the sample mean is between 7 hours and 9 hours?
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:

Problem 03 Normal Distribution: Paranoid Schizophrenia In a study of facial behavior, people in a control group [100 subjects randomly selected from general population] are timed for eye contact in a 5-minute period. Their times are normally distributed with a mean of 188.0 seconds and a standard deviation of 58.0 seconds (based on data from “Ethological Study of Facial Behavior in Nonparanoid and Paranoid Schizophrenic Patients,” by Pittman, Olk, Orr, and Singh, Psychiatry, Vol. 144, No. 1).

(a) For a randomly selected person from the control group, find the probability that the eye contact time is greater than 230.0 seconds, which is the mean for paranoid schizophrenics.
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:

(b) Based on personal experience, do you believe the article’s result based on eye contact is a valid indicator for paranoid schizophrenics?
Problem 4 Medical: White Blood Cells Let x be a random variable that represents white blood cell count per cubic milliliter of whole blood. Assume that x has a distribution that is
normal with a mean μ = 7,580 and a standard deviation σ = 2,530. A test result of x < 3,500 is an indication of leukopenia. An emergency room patient is given the following sequence of tests to determine whether leukopenia is possible diagnosis. Emergency
treatment can have many detrimental side effects, so the ER staff must be certain that the test results are an accurate predictor of leukopenia.
(a) What is the probability that, on a single test, x is less than 3,500?
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:

(b) Suppose the staff uses the average xbar for two tests taken about 1 hour apart. What is the probability of xbar < 3,500?
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:
(c) Suppose the staff uses the average xbar for three tests taken about 1 hour apart. What is the probability of xbar < 3,500?
Calculator Function & Inputs:
(d) The staff uses the average xbar for four tests taken about 1 hour apart. What is the probability of xbar < 3,500?
Calculator Function & Inputs:
(e) Which test sequence would you have the most confidence in determining whether leukopenia is possible diagnosis.?
(a) (b) (c) (d)
Why?

Problem 5 Normal Distribution: Navy Pilots The U.S. Navy requires that its pilots have heights between 61 in. and 77 in. Given the following population parameters:

Men’s heights are normally distributed with μ = 72.8 in. and σ = 2.7 in. Women’s heights are normally distributed with μ = 63.8 in. and σ = 2.9 in.

If the height requirements are changed to exclude the tallest 1% of the men and the shortest 4% of the women, what are the new height requirements?
𝑴𝒆𝒏: Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:

𝑾𝒐𝒎𝒆𝒏: Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:
New Height Requirements:

Problem 6 Normal Distribution: Ambulance Response Time Ambulance response time in Edmonton, Alberta is measured as the time (in minutes) between the initial call to emergency medical services (EMS) and when the patient is reached by ambulance. For a particular EMS station (call it Station A), ambulance response time is known to be normally distributed with μ = 7.5 minutes and σ = 2.5 minutes. A randomly selected EMS call in Edmonton has an ambulance response time of 2 minutes. Is it likely that this call was serviced by Station A? Explain.
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:
Problem answer:( Yes / No )
Why?

Problem 7 Normal Distribution: Accrotine Watches Accrotime is a manufacturer of underwater quartz crystal watches. Its researchers have shown that the watches have an average life μ = 49 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is σ = 9 months and the population distribution of lifetimes is normal.

(a) If Accrotime guarantees a full refund on any defective watch for 3 years after purchase, what percentage
of total production should the company expect to replace?
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:

(b) If Accrotime does not want to make refunds on more than 3% of the watches it makes, how long should the guarantee period be?
Guiding Statement: Given , find !
Probability Statement:
Calculator Function & Inputs:

## Solution Preview

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Problem 01: Normal Distribution: Test Times The length of time, x, it takes to finish a statistics progress quiz is normally distributed with a mean of 18.0 minutes and a standard deviation of 1.3 minutes. The time limit for the quiz is set at 20 minutes. For a class of 38 students, how many students will finish the test?
Guiding Statement: Given x ~ N(18, 1.3), find 38 × P(x < 20)
Probability Statement: P(x < 20)
Calculator Function & Inputs: normalcdf(-1E99, 20, 18, 1.3)
Number of students: show your calculation 38 × 0.938032 = 35.65 or 36 (rounded)
Problem 02 Sampling Distributions: Sleep Studies The population of the amount of time that an adult sleeps is normally distributed with a mean of 6.8 hours and a standard deviation of 1.4 hours (based on data from multiple sources, including a Gallup poll and the American Journal of Epidemiology). A common recommendation is that we get between 7 and 9 hours of sleep each night.
(a) For someone randomly selected, find the probability that he/she gets between 7 and 9 hours of sleep in a night.
Guiding Statement: Given x ~ N(6.8, 1.4), find P(7 < x < 9)
Probability Statement: P(7 < x < 9)
Calculator Function & Inputs: normalcdf(7, 9, 6.8, 1.4)
Calculator Answer: 0.3852 (rounded to 4 decimal places)
(b) If we randomly collect a sample of 5 adults, what is the probability that the sample mean is between 7 hours and 9 hours?
Guiding Statement: Given x̅~N(6.8, 1.4/√5), find P(7 < x̅ < 9)
Probability Statement: P(7 < x̅ < 9)...

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