Statistics 160 — 040
Show all your work on the pages of this examination paper. Use the back of the pages if sumcient space is not available.
(10) I. The game of Yahtzee involves rolling 5 die simultaneously. Points are awarded for obtaining various combinations, such as 4 of a kind or a straight (5 consecutive numbers 1 to 5 or 2 to 6). For the purpose of computing probabilities it may be easier to think of five blanksand fill each blank with a random number from I to 6. Calculate the following probabilities.
- 5 distinct numbers (i.e. no two dice showing the same number).
- 5 of a kind (called a yahtzee).
(c) 4 of a kind.
c; .6-5 5.5 25
(d) Full house (3 of a kind and 2 of a kind).
/o.6-5 50 25
(8) 2. Five hundred employees of some private companies were randomly selected and asked whether or not they have any retirement benefits provided by their companies:
|Have Retirement Benefits||Do Not Have Retirement Benefits|
375 tas 500
- If an employee is randomly selected from this sample, find the probability that the employee has retirement benefits.
- If an employee is randomly selected from this sample, find the probability that the employee is a women who does not have retirement benefits.
- Are the events “Employee is Male” and “Employee has Retirement Benefits” independent? You must calculate the appropriate probabilities as part of your
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500 goo aas p (M R) M R a-cc
(6) 3. A study conducted by a bicycle distributor showed the number of bicycles owned per teenager and the corresponding probabilities for each. Find the mean (expected number of bicycles per teenager) and the standard deviation for this distribution.
|Number of bicycles (x)||o||1||2||3|
|Probability P (x)||0.18||0.55||0.23||0.04|
18) + I 55) + a 03) +
– o. 71/37
(12) 4. The table below gives data for a random sample of eight students showing the amount of outside class study time spent and the grade earned at the end of a one month statistics course.
|Study Time (hr) x||20||16||34||23||27||32||18||22|
Part of a linear regression analysis has been completed with the following sums already calculated:
= 192, = 4902, Ey = 608, Ey2 = 47094, = 15032.
(a) Find the following values: sx, sy, smy, and the correlation coefficient r for the above data. Show the formulas and calculation.
. 86 a
- Find the equation y = a + bm of the linear regression (best fit) line. Show the formulas and calculation.
- What is the predicted grade for a student who studied 25 hours?
+0.084 1. 477 (as) 77.51
(10) 5. Assume that 8% of the skunks in Saskatchewan have rabies.
(a) If twenty skunks are selected at random, what is the probability that exactly one of them has rabies?
pc k -l) CIO3282
(b) If 200 were selected at random, use the normal distribution (with continuity correction) to estimate the probability that less than 15 have rabies. = 3.33667
15) = pa </+.5) pe < 1+.5—/6
3.83667) -.370%65) . 31B 7 %
P(k< 15) = S GOO (.03) (.72) .3597
(8) 6. Suppose that the life span of a new calculator is normally distributed with mean 54.4 months and standard deviation 8.2 months.
- The company guarantees that any calculator that starts malfunctioning within 36 months of the purchase will be replaced by a new one. What percentage of calculators manufactured by this company are expected to be replaced?
- (2<36) PG z 36-5%.f8.2 ) p (z z —a. 2+37)
- A random sample of 45 calculators is selected. What is the probability that the average lifespan of the calculators in this sample will be greater than 54 months?
- (i > 54) — p (z > 54—8 a/1/V5 p (z > 7a3) P (z < .32 723) . can
(12) 7. A glass marble manufacturer claims that their marbles have a diameter of 12 mm. A sample of 100 of their marbles was randomly selected. The sample mean was 12.2 mm and the sample standard deviation was 0.7 mm.
- State the null and alternate hypotheses. Give the test statistic and with a = .05 use it to test the manufacturer’s claim.
IQ 74 IQ
- Construct a 95% confidence interval for estimating the true mean diameter of their marbles.
(12.0628 la. 3370
(c) Does the confidence interval in part (b) support the claim in part (a)? Explain.
(10) 8. Two random samples of people in two different age groups (Population A: above 60 years old, and Population B: below 40 years old) who live in Moose Jaw were taken to determine the proportion of people that sleep less than 8 hours per night. A sample of size 250 is taken from Population A, and 150 of them sleep less than 8 hours per night. A second sample of size 250 is taken from Population B, and 172 of them sleep less than 8 hours per night. At a = .05 , does the data demonstrate that the proportion of persons who sleep less that 8 hours per night is significantly lower for Population A than that for Population B?
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(7) 9. A compact disc manufacturer claims that their discs have a mean diameter of 120mm. A rumor is circulating around that their discs are undersized. To dispel the rumor a sample of a dozen of their discs is randomly selected. The sample mean was 119.56 mm and the sample standard deviation was 0.90 mm. State the null and alternate hypotheses. At 95% confidence is there enough evidence to support the rumor? What must we assume about the distribution of the diameters of their discs?
(7) 10. A drug manufacturer selected a sample of 30 of their sleeping pills for testing. The sample had an average of 25.1 mg of active ingredient per pill, with a standard deviation of 0.52 mg. Construct a 90% confidence intel•val for the true standard deviation of the active ingredient in their sleeping pills.
c < . 6655
(10) 11. We suspect that a die is loaded so that the number 6 comes up more often than expected. An experiment involving rolling the die 60 times is conducted. Here are the frequencies of each number showing up:
- Use the Pearson X2 test with a = .05 to determine if the die is loaded. Show your work and clearly state your conclusion.
- If we were to roll the die 120 times and the same relative frequencies were to show up would you change your conclusion? Show your work.