Normal Probability Distribution — practice test 6B

Jan 12, 2017

Problems with solutions in Elementary Statistics — The Standard Normal Distribution, Applications of Normal Distributions, Sampling Distributions and Estimators, The Central Limit Theorem, Normal as Approximation to Binomial, Assessing Normality

Directions:
Give your answers to the short-answer items. Choose the correct answer for multiple-choice items.

Provide an appropriate response.
- 1. Complete the following table for a distribution in which μ = 16. It might be helpful to make a diagram to help you determine the continuity factor for each entry.
  
  Possible answer
Using the following uniform density curve, answer the question.
- 2.
  What is the probability that the random variable has a value between 2.6 and 5.4?
  1. 0.2250
  2. 0.3500
  3. 0.6000
  4. 0.4750
Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost.
- 3. Less than 10 pounds
  1. ⁵/₇
  2. ¹/₃
  3. ²/₃
  4. ¹/₆
If Z is a standard normal variable, find the probability.
- 4. The probability that Z lies between –0.55 and 0.55
  1. –0.9000
  2. –0.4176
  3. 0.9000
  4. 0.4176
- 5. P(Z < 0.59)
  1. 0.7224
  2. 0.2776
  3. 0.2190
  4. 0.2224
The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information.
- 6. If 9% of the thermometers are rejected because they have readings that are too high, but all other thermometers are acceptable, find the temperature that separates the rejected thermometers from the others.
  1. 1.45°
  2. 1.26°
  3. 1.39°
  4. 1.34°
- 7. If 6.3% of the thermometers are rejected because they have readings that are too high and another 6.3% are rejected because they have readings that are too low, find the two readings that are cutoff values separating the rejected thermometers from others.
  1. –1.53°, 1.53°
  2. –1.45°, 1.45°
  3. –1.46°, 1.46°
  4. –1.39°, 1.39°
Assume that X has a normal distribution, and find the indicated probability.
- 8. The mean is μ = 15.2 and the standard deviation is σ = 0.9.
  Find the probability that X is greater than 15.2.
  1. 0.0003
  2. 1.0000
  3. 0.9998
  4. 0.5000
- 9. The mean is μ = 22.0 and the standard deviation is σ = 2.4.
  Find the probability that X is between 19.7 and 25.3.
  1. 1.0847
  2. 0.3370
  3. 0.7477
  4. 0.4107
Solve the problem.
- 10. Suppose that replacement times for washing machines are normally distributed with a mean of 8.7 years and a standard deviation of 1.6 years. Find the replacement time that separates the top 18% from the bottom 82%.
  1. 10.2 years
  2. 7.2 years
  3. 9.6 years
  4. 9.0 years
Find the indicated probability.
- 11. The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3oz and a standard deviation of 1.2oz. What is the probability that the volume of soda is a randomly selected bottle will be less than 32oz?
  1. 0.3821
  2. 0.0987
  3. 0.5987
  4. 0.4013
Provide an appropriate response.
- 12. A recent survey based on a random sample of n = 490 voters, predicted that the Independent candidate for the mayoral election will get 23% of the vote, but he actually get 28%. Can it be concluded that the survey was done incorrectly?
  
  Possible answer
  No, because of sampling variability, sample proportions will naturally vary from the true population proportion, even if sampling is done with a perfectly valid procedure.
List the different possible samples, and find the mean of each of them.
- 13. The number of books sold over the course of the four-day book fair were 156, 200, 293, and 55. Assume that samples of size 2 are randomly selected with replacement from this population of four values.
  
  Possible answer
  Possible samples: 156–156; 156–200; 156–293; 156–55; 200–156; 200–200; 200–293; 200–55; 293–156; 293–200; 293–293; 293–55; 55–156; 55–200; 55–293; 55–55
  Means: 156, 178, 224.5, 105.5, 178, 200, 246.5, 127.5, 224.5, 246.5, 293, 174, 105.5, 127.5, 174, 55
Solve the problem.
- 14. The weights of the fish in a certain lake are normally distributed with a mean of 15 lb and a standard deviation of 6. If 4 fish are randomly selected, what is the probability that the mean weight will be between 12.6 and 18.6 lb?
  1. 0.0968
  2. 0.4032
  3. 0.6730
  4. 0.3270
- 15. A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 8.7 hours.
  1. 0.1469
  2. 0.1285
  3. 0.1946
  4. 0.1346
Use the continuity correction and describe the region of the normal curve that corresponds to the indicated binomial probability.
- 16. The probability of fewer than 56 democrats
  1. The area to the left of 55.5
  2. The area to the right of 56.5
  3. The area to the left of 56
  4. The area to the left of 56.5
For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation.
- 17. n = 15, p = .5
  1. Normal approximation is suitable.
  2. Normal approximation is not suitable.
Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
- 18. A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is correct. If all answers are random guesses, estimate the probability of getting at least 20% correct.
  1. 0.0901
  2. 0.1492
  3. 0.8508
  4. 0.3508
Solve the problem.
- 19. A normal probability plot is given below for the lifetimes (in hours) of a sample of batteries of a particular brand. Use the plot to assess the normality of the lifetimes of these batteries. Explain your reasoning.
  
  Possible answer
  Since the normal probability plot displays curvature, it appears that lifetimes of these batteries are probably not normally distributed.
Examine the given data set and determine whether the requirement of a normal distribution is satisfied. Assume that the requirement for a normal distribution is loose in the sense that the population distribution need not be exactly normal, but it must have a distribution which is basically symmetric with only one mode. Explain why you do or do not think that the requirement is satisfied.
- 20. The heart rates (in beats per minute) of 30 randomly selected students are given below.
  
  Possible answer
  The requirement for normality is satisfied since a histogram of the data is roughly bell shaped; it is roughly symmetric with a single mode.

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