Probability - normal distribution, including normal approximation to binomial

1. One of the side effects of flooding a lake in northern boreal forest areas (e.g. for a hydro-electric project) is that mercury is leached from the soil, enters the food chain, and eventually contaminates the fish. The concentration in fish will vary among individual fish because of differences in eating patterns, movements around the lake, etc. Suppose that the concentrations of mercury in individual fish follows an approximate normal distribution with a mean of 0.25 ppm and a standard deviation of 0.08 ppm. Fish are safe to eat if the mercury level is below 0.30 ppm. What proportion of fish are safe to eat?
   1. 63%
   2. 23%
   3. 73%
   4. 27%
   5. 37%
   Solution c Past performance - 92 Dec - 45% (16%a, 22%b, 15%d) Past performance - 93 Apr - 57% (a-17%; d-17%) Past performance - 96 Nov - 93% Past performance - 97 Aug - 84%
2. Refer to the previous question. The Department of Fisheries and Oceans wishes to know the mercury level of the top 20% of the fish. The appropriate percentile and mercury level for this lake is:
   1. 20th percentile has a value of -0.84 ppm
   2. 20th percentile has a value of 0.18 ppm
   3. 80th percentile has a value of 0.32 ppm
   4. 80th percentile has a value of 0.84 ppm
   5. 20th percentile has a value of 0.07 ppm
   Solution c Past performance - 92 Dec - 46% (28%-b, 15%-d) Past performance - 97 Aug - 77% (13%-d)
3. The following graph is a normal probability plot for the amount of rainfall in acre-feet obtained from 26 randomly selected clouds that were seeded with silver oxide:
   
   
   1. The data appear to show exponential growth; that is, the amount of rainfall increases exponentially as the amount of silver oxide increases.
   2. The pattern suggests that the measurement is not normally distributed.
   3. A least squares regression line should be fitted to the rainfall variable.
   4. It can be expected that the histogram of rainfall amount will look like the normal curve.
   5. The shape of the curve suggests that rainfall is caused by seeding the clouds with silver oxide.
   Solution not available
4. Marks on a Chemistry test follow a normal distribution with a mean of 65 and a standard deviation of 12. Approximately what percentage of the students have scores below 50?
   1. 11%
   2. 89%
   3. 15%
   4. 18%
   5. 39%
   Solution a
5. Refer to the preceding question. What is the approximate 90th percentile of the mark distribution?
   1. 80
   2. 90
   3. 85
   4. 75
   5. 95
   Solution a
6. The marks on a statistics test are normally distributed with a mean of 62 and a variance of 225. If the instructor wishes to assign B's or higher to the top 30% of the students in the class, what mark is required to get a B or higher?
   1. 68.7
   2. 71.5
   3. 73.2
   4. 74.6
   5. 69.9
   Solution e Past performance - 89 Dec - 50% (25% -d, 10% -b,c) Past performance - 91 Oct - 67% (10% c, 14% d)
7. The grade point averages of students at the University of Manitoba are approximately normally distributed with mean equal to 2.4 and standard deviation equal to 0.8. What fraction of the students will possess a grade point average in excess of 3.0 ?
   1. .7500
   2. .6000
   3. .2734
   4. .2500
   5. .2266
   Solution e Past performance - 89 Dec - 52% (18% c,d) Past performance - 89 Apr - 50% (C-23%, D-18%) Past performance - 91 Dec - 80% (c-13%)
8. In some courses (but certainly not in an intro stats course!), students are graded on a "normal curve". For example, students within ± 0.5 standard deviations of the mean receive a C; between 0.5 and 1.0 standard deviations above the mean receive a C+; between 1.0 and 1.5 standard deviations above the mean receive a B; between 1.5 and 2.0 standard deviations above the mean receive a B+, etc. The class average in an exam was 60 with a standard deviation of 10. The bounds for a B grade and the percentage of students who will receive a B grade if the marks are actually normal distributed are:
   1. (65, 75), 24.17%
   2. (70, 75), 18.38%
   3. (70, 75), 9.19%
   4. (65, 75), 12.08%
   5. (70, 75), 6.68%
   Solution c Past performance - 97 Jul - 85%

   Refer to the previous question. Another Instructor decides that the lower "B" cutoff should be the 70th percentile. The lower-cutoff for a B grade is:

   1. 70
   2. 65
   3. 60
   4. 75
   5. 80
   Solution b Past performance - 97 Jul - 71% (14%-a)
9. The diameters of steel disks produced in a plant are normally distributed with a mean of 2.5 cm and standard deviation of .02 cm. The probability that a disk picked at random has a diameter greater than 2.54 cm is about:
   1. .5080
   2. .2000
   3. .1587
   4. .0228
   5. .4920
   Solution d
10. Suppose the test scores of 600 students are normally distributed with a mean of 76 and standard deviation of 8. The number of students scoring between 70 and 82 is:
    1. 272
    2. 164
    3. 260
    4. 136
    5. 328
    Solution e
11. Bolts that are used in the construction of an electric transformer are supposed to be 0.060 inches in diameter, and any bolt with diameter less than 0.058 inches or greater than 0.062 inches must be scrapped. The machine that makes these bolts is set to produce bolts of 0.060 inches in diameter, but it actually produces bolts with diameters following a normal distribution with µ=0.060 inches and =0.001 inches. The proportion of bolts that must be scrapped is equal to:
    1. 0.0456
    2. 0.0228
    3. 0.9772
    4. 0.3333
    5. 0.1667
    Solution a
12. The cost of treatment per patient for a certain medical problem was modeled by one insurance company as a normal random variable with mean $775 and standard deviation $150. What is the probability that the treatment cost of a patient is less than $1,000, based on this model?
    1. .5000
    2. .6826
    3. .8531
    4. .9332
    5. Cannot be computed without knowledge of additional parameters
    Solution d
13. The time that a skier takes on a downhill course has a normal distribution with a mean of 12.3 minutes and standard deviation of 0.4 minutes. The probability that on a random run the skier takes between 12.1 and 12.5 minutes is:
    1. 0.1915
    2. 0.3830
    3. 0.3085
    4. 0.6170
    5. 0.6826
    Solution b
14. It is known that the resistance of carbon resistors is normally distributed with µ=1200 ohms and = 120 ohms. What proportion of the resistors have resistances that differ from the mean resistance by more than 120 ohms?
    1. 0.9544
    2. 0.3413
    3. 0.1587
    4. 0.6826
    5. 0.3174
    Solution e
15. The time required to assemble an electronic component is normally distributed with a mean of 12 minutes and a standard deviation of 1.5 min. Find the probability that a particular assembly takes more than 14.25 minutes.
    1. .9332
    2. .0668
    3. .3413
    4. .4332
    5. .1587
    Solution b
16. Heights of males are approximately normally distributed with a mean of 170 cm and a standard deviation of 8 cm. What fraction of males are taller than 176 cm?
    1. .7500
    2. .6000
    3. .2734
    4. .2500
    5. .2266
    Solution e Past performance - 90 Oct - 68% Past performance - 93 Feb - 87%
17. The height of an adult male is known to be normally distributed with mean of 175 cm and standard deviation 6 cm. The 20th percentile of the distribution of heights is:
    1. 175
    2. 179
    3. 170
    4. 172
    5. 174
    Solution c
18. The heights of students at a college are normally distributed with a mean of 175 cm and a standard deviation of 6 cm. One might expect in a sample of 1000 students that the number with heights less than 163 cm is:
    1. 997
    2. 23
    3. 477
    4. 228
    5. 456
    Solution b Past performance - 91 Oct - 62% (12% c, 20% d) Past performance - 96 Dec - 83% (11% d)
19. The height of an adult male is known to be normally distributed with a mean of 69 inches and a standard deviation of 2.5 inches. The height of the doorway such that 96 percent of the adult males can pass through it without having to bend is:
    1. 1.75
    2. 64.63
    3. 73.38
    4. 79.94
    5. 58.06
    Solution c
20. The distribution of weights in a large group is approximately normally distributed. The mean is 80 kg. and approximately 68% of the weights are between 70 and 90 kg. The standard deviation of the distribution of weights is equal to:
    1. 20
    2. 5
    3. 40
    4. 50
    5. 10
    Solution e
21. The distribution of weights of a large group of high school students is normally distributed with µ=55 kg and =5 kg. Which of the following is true?
    1. About 16 percent of the students will be over 60 kg.
    2. About 2.5 percent will be below 45 kg.
    3. Half of them can be expected to weigh less than 55 kg.
    4. About 5 percent will weigh more than 63 kg.
    5. All the above are true.
    Solution e
22. The daily milk production of Guernsey cows is approximately normally distributed with a mean of 35 kg/day and a std. deviation of 6 kg/day. The probability that a days production for a single animal will be less than 28 kg. is approximately:
    1. .41
    2. .09
    3. .38
    4. .12
    5. .62
    Solution d Past performance - 90 Dec - 66%
23. Refer to the previous question. The producer is concerned when the milk production of a cow falls below the 5th percentile since the animal may be ill. The 5th percentile (in kg) of the daily milk production is approximately:
    1. 1.645
    2. -1.645
    3. 33.36
    4. 25.13
    5. 44.87
    Solution d Past performance - 90 Dec - 64%
24. Which of the following is NOT CORRECT about a standard normal distribution?
    1. P( 0 < Z < 1.50) = .4332
    2. P( Z < -1.0 ) = .1587
    3. P( Z > 2.0 ) = .0228
    4. P( Z < 1.5 ) = .9332
    5. P( Z > -2.5) = .4938
    Solution e Past performance - 89 Dec - 78% Past performance - 90 Oct - 76%
25. The measurement of the width of the index finger of a human right hand is a normally distributed variable with a mean of 6 cm. and a standard deviation of 0.5 cm. What is the probability that the finger width of a randomly selected person will be between 5 cm. and 7.5 cm.?
    1. .9759
    2. .0241
    3. .9500
    4. 1.000
    5. not within ± 0.001 of these
    Solution a

    Normal approx. to binomial

26. The National Broomball League claims to have a balanced league; that is, for any given game each team has an equal chance of winning or losing with no ties. Assuming the claim is true, what is the approximate probability that a given team will lose more than 61 games out of the 100 played?
    1. 0.0500
    2. 0.4918
    3. 0.0107
    4. 0.0082
    5. 0.0164
    Solution not available
27. The probability of getting a parking ticket when not paying for a 2-hour period is 0.3. What is the probability of getting at least 60 tickets if you park on 250 occasions for a 2-hour period and don't pay?
    1. 0.016
    2. 0.019
    3. 0.98
    4. 0.93
    5. 0.072
    Solution not available
28. A professional basketball player sinks 80% of his foul shots, in the long run. If he gets 100 tries during a season, then the probability that he sinks between 75 and 90 shots (inclusive) is approximately equal to:
    1. Pr ( -1.25 <= Z <= 2.5 )
    2. Pr ( -1.125 <= Z <= 2.625 )
    3. Pr (-1.125 <= Z <= 2.375 )
    4. Pr ( -1.375 <= Z <= 2.375 )
    5. Pr (-1.375 <= Z <= 2.625 )
    Solution not available
29. Suppose in the University of Manitoba, 30% of the students live in apartments. If 200 students are randomly selected, then the probability that the number of them living in apartments will be between 50 and 75 inclusive, is:
    1. .9167
    2. .9298
    3. .9390
    4. .9268
    5. .9208
    Solution not available
30. If X has a binomial distribution with n = 400 and p = .4, the approximate probability of the event {155 < X < 175} is:
    1. 0.6552
    2. 0.6429
    3. 0.6078
    4. 0.6201
    5. 0.6320
    Solution not available
31. If in the previous question we change the interval to 155 <= X <= 175, the approximate probability is;
    1. 0.4
    2. larger than that in the previous question
    3. smaller than that in the previous question
    4. equal to that in the previous question
    5. may be smaller or larger than that in the previous question
    Solution not available
32. Companies are interested in the demographics of those who listen to the radio programs they sponsor. A radio station has determined that only 20% of listeners phoning in to a morning talk program are male. During a particular week, 200 calls are received by this program. What is the approximate probability that at least 50 of the callers are male?
    1. .0466
    2. .0212
    3. .1168
    4. .1402
    5. Not within ± .01 of any of the above.
    Solution not available
33. The unemployment rate in a certain city is 8.5%. A random sample of 100 people from the labour force is drawn. Find the approximate probability that the sample contains at least ten unemployed people.
    1. .3879
    2. .3245
    3. .3419
    4. .2946
    5. .3594
    Solution not available
34. A politician has targeted 100 homes to visit during a week. From past experience, 50 percent of the households answer the bell and invite him in. Of this, 80 percent will agree with his policies. The approximate probability that the politician will get support from at least 45 households during a week is:
    1. 0.1991
    2. 0.3212
    3. 0.8643
    4. 0.1376
    5. 0.1788
    Solution not available
35. People who have been in contact with a carrier of a disease, have a 40% chance of contracting the disease. Suppose that the carrier of the diseases may have infected a school with 500 people. Find the approximate probability that at least 215 people will contract the disease.
    1. .09
    2. .91
    3. between .05 and .34
    4. 1.37
    5. between 2.5% and 17%
    Solution a Past performance - 93 Apr - 40% (b-22%, c-22%)

These problems were captured from the Simon Fraser University web site, and moved here for efficiency. We are greatful to the original authors.