1) Suppose that for a certain distribution the mean is 64.2 and the standard deviation is 7.8. Between which 2 values would at least 84% of the distribution lie? (44.7 and 83.7)
2) How many standard deviations away from the mean are 31.05 and 97.35? What is the minimum percentage of the distribution between these limits? (k = 4.25; 94.46%)
3) A market research firm has a standard monthly customer satisfaction survey for one of its clients. It tracks the time needed by the interviewers to complete the survey which follows this distribution (in minutes rounding to the nearest 15 seconds):
|
x |
2 |
2.25 |
2.5 |
2.75 |
3 |
3.25 |
3.5 |
|
p(x) |
0.1 |
0.25 |
0.43 |
0.11 |
0.08 |
0.02 |
0.01 |
In theory, completion times should be within 2 standard deviations of the mean. Based on Chebyshev’s theorem using k=2, for which completion times should employees receive additional training? (Those over 3 minutes)
4) In a wine cork manufacturing plant, the width of the corks should be between 21.991 mm and 22.009 mm. For quality control purposes, 6 corks are sampled each minute from a production of 24. If at least 1 of the corks is outside the limits, the machine is adjusted. Suppose that a group of 24 corks has 3 that are outside the limits. What is the probability the machine will be adjusted? (59.68%)
5) Suppose instead that a group of 48 corks has 6 that are outside the limits. From a quality control perspective, would it be better to sample 12 every 2 minutes from a production of 48 and adjust the machine if at least 1 cork is outside the limits? (Yes, since the probability of adjusting the machine is now 84.13%)
6) Sara is a telemarketer who makes 20 calls per day. Her average success rate is 5%. What is the probability she will make exactly 2 sales per day? (18.87%)
7) What is the probability that on any given day she will make no more than 1 sale? (73.59%)
8) Suppose her goal is to make at least 2 sales per day. What is the probability she will do that? (26.41%)
9) Suppose she earns a commission of $450 per sale. If she works 5 days a week and makes the same number of calls each day, what is the maximum number of sales she can expect to make 99.7% of the time, rounding to the nearest whole number? Based on that, what is the maximum commission she can expect to make? ($5400)
10) In an automobile manufacturing plant, approximately 1 gear in 5000 is scrapped. If the plant produces 500 gears per shift, what is the probability that none of the gears is scrap? (90.48%)
11) The plant has 3 shifts per day. What is the probability that in this time frame that at least 1 gear is scrap? (25.92%)
12) The plant has a quality control goal of having no more than 6 scrap gears in a 7-day period. What is the probability of this? Is this a realistic goal? (99.42%; it seems realistic)
13) Using Chebyshev’s theorem, what is the maximum number of scrap gears they can expect in a 7-day period at least 88.9% of the time, rounding to the nearest whole number? Why is Chebyshev’s probability conservative? (6 gears; as we see in the previous question, the probability of no more than 6 scrap gears is 99.42% which is considerably higher than the 88.9% under Chebyshev’s theorem.)
14) In the same plant, there is 1 accident per 60 days on average. What is the probability the plant has no accidents in a 30-day period? (60.65%)
15) What is the probability that there are at least 2 accidents in a 180-day period? (80.09%)
16) Using Chebyshev’s theorem, what is the maximum number of accidents they can expect in a 360-day period at least 88.9% of the time, rounding to the nearest whole number? What is the probability the plant has no more than this many accidents in this time frame? (13 accidents; 99.64%)