Stats 3Y03 Sample Test #3
Question Sheet
Name:___________________________________ _____________________Student Number:
(Last Name) (First Name)
All solutions must be written on the one page answer sheet. Additional pages will not be marked. This test consists of 7 long answer questions (full solutions required), worth 5 marks each.
You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Only the McMaster standard calculator Casio FX-991 is allowed.
Question #1: (a) Suppose that is the number of observed "successes" in a sample of \ 8
observations where is the probability of success on each observation and the:
observations are independent. Suppose that we use s: œ# \8## as an estimator of :#. Find the amount of bias in the estimator s:#.
(b) Two different plasma etchers in a semiconductor factory have the same mean etch rate . However, machine 1 is newer than machine 2 and consequently has.
smaller variability in etch rate. We know that the variance of etch rate for machine 1 is 5"# and for machine 2 is 5## "5"#. Suppose that we have a sample
$
œ
of 8" independent observations on etch rate from machine 1 and 8# independent
observations on etch rate from machine 2, and that the samples are independent. Find the bias and standard error of the estimator .s œ \ \"$ " #$ #.
Question #2: (a) A medical researcher wishes to estimate the percentage of females who take vitamins. He wishes to be 98% confident that the estimate is within 4 percentage points of the true proportion. What is the minimum sample size needed?
(b) In order to estimate the average age of onset of a certain type of disease a researcher collects a sample of 15 people with the disease and produces the following 95% confidence interval (83.9219, 90.6781). Find a 99% confidence interval for the average age based on the same data set.
(c) In order to estimate the proportion of people who are under the age of 21 that enter a particular hospital emergency room each week a researcher takes a sample and produces the following 90% confidence interval for the proportion of people that are under the age of 21 (0.095970, 0.304030). What sample size was used?
Question #3: An article in Food Testing and Analysis "Improving Reproducability of Refractometry Measurements of Fruit Juices" (1999, Vol. 4, No. 4, pp. 13-17) measured the sugar concentration (Brix) in clear apple juice. All measurements were taken at 20°C:
""Þ%)ß ""Þ%&ß ""Þ%)ß ""Þ%(ß ""Þ%)ß ""Þ&!ß ""Þ%#ß ""Þ%*
(a) Test the hypothesis L À! .œ ""Þ& versus L À" .Á ""Þ& using αœ Þ!&.
(b) Based on your conclusion in (a), is it possible that a Type II error has occurred? Explain.
(c) Find the -value.:
(d) What assumptions (if any) are required for the above test?
(e) What method could be used to check the assumption in (c)?
Question #4: During the 1999 and 2000 baseball seasons, there was much speculation that the unusually large number of home runs there were hit was due at least in part to a livelier ball. One way to test the "liveliness" of a baseball is to launch the ball at a vertical surface with a known velocity ZP and measure the ratio of the outgoing velocity ZS of the ball to ZP. The ratio V œ Z ÎZS P is called the coefficient of restitution. A batch of 40 baseballs were tested by throwing each ball from a pitching machine at an oak surface. The average coefficient of restitution was found to be B œ Þ'$($, with standard deviation = œ Þ!"$. If the mean coefficient of restitution exceeds .635, then the population of balls from which the sample was taken will be too "lively", and considered unacceptable for play.
(a) Test the hypothesis that the balls in the sampled population are too lively using the 5% significance level.
(b) Find the -value.:
(c) Find the power of the hypothesis test in (a) if the true mean coefficient of restitution is equal to .638
(b) What sample size would be required to detect a true mean coefficient of restitution of .638 if we want the power of the test to be at least 0.90?
Question #5: A university library ordinarily has a complete shelf inventory done once every year. Because of new shelving rules instituted the previous year, the head librarian believes it may be possible to save money by postponing the inventory. The librarian decides to select at random 1000 books from the library's collection and have them searched in a preliminary manner. If evidence indicates strongly that the true proportion of misshelved or unlocatable books is less than .02, then the inventory will be postponed.
(a) Among 1000 books searched, 15 were misshelved or unlocatable. Test the relevant hypothesis and advise the librarian what to do (use αœ Þ!&Ñ.
(b) Find the -value.:
Question #6: A person wants to test whether a die is unbalanced. He thinks that the die has been weighted so that "6"'s appear more often than the other numbers. So he wants to test
L À : œ " L À : "
' '
! versus "
where is the probability of rolling a "6". He decides to roll the die 8 times, and:
he will reject L! is he observes 3 or more "6"'s.
(a) What is the significance level of this test?
(b) Find the power of the test if the true value of is .: "%
Question #7: The "spring-like" effect in a golf club could be determined by measuring the coefficient of restitution (the ratio of the outbound velocity to the inbound velocity of a golf ball fired at the club head). Drivers are randomly selected from two club makers and the coefficient of restitution is measured. The data are as follows, and are summarized in the Minitab output below:
Club 1:0.8906 0.8104 0.8234 0.8198 0.8235 0.8562 0.8123 0.7976 0.8184 ß ß ß ß ß ß ß ß ß
0.8265 0.7173 0.7871ß ß ß
Club 2:0.8305 0.7905 0.8352 0.8380 0.8145 0.8465 0.8244 0.8014 0.8309ß ß ß ß ß ß ß ß ß
0.8405
---Test for Equal Variances: C1 versus C2
95% Bonferroni confidence intervals for standard deviations
C2 N Lower StDev Upper 1 12 0.0276390 0.0408729 0.0755540 2 10 0.0117375 0.0179439 0.0361327
F-Test (normal distribution)
Test statistic = 5.19, p-value = 0.020
Levene's Test (any continuous distribution) Test statistic = 1.19, p-value = 0.289
Descriptive Statistics: C1
Variable C2 N N* Mean SE Mean StDev Minimum Q1 Median C1 1 12 0 0.8153 0.0118 0.0409 0.7173 0.8008 0.8191 2 10 0 0.82524 0.00567 0.01794 0.79050 0.81123 0.83070
Variable C2 Q3 Maximum C1 1 0.8258 0.8906 2 0.83863 0.84650
---(a) Is an assumption of equal variances justified?
(b) Test the hypothesis that the clubs from brand 2 have a greater mean coefficient of restitution than the clubs from brand 1. Use αœ Þ!&.
(c) Find the -value for the test in : (b).
Stats 3Y03 Sample Test #3
Answer Sheet (additional pages will not be marked)
Name:___________________________________ _____________________Student Number:
(Last Name) (First Name)
____________________________________________________________________________ Question #1:
____________________________________________________________________________ Question #2:
(additional pages will not be marked)
____________________________________________________________________________ Question #4:
____________________________________________________________________________ Question #5:
____________________________________________________________________________ Question #6:
Answers for Sample Test #3 1. (a) :Ð":Ñ8 (b) 0, 5$" 8" $8%
" #
2. (a) 849 (b) (82.61 91.99) ß (c) 40 (d) 93%
3. (a) Reject L! since $Þ#" #Þ$'& (b) No.
(c) Þ!" T Þ!#
(d) The population must be normal
(e) Normal probability plot
4. (a) Do not reject L! since 1.119 is less than 1.64
(b) 0.131
(c) 0.426
(d) 161
5. (a) Do not reject L! since "Þ"$ is not less than "Þ'%. The inventory should not be postponed
(b) .129
(c) .192
6. (a) .135
(b) .321
(c) No.
7. (a) no, since the -value of .02 is less than .05:
(b) Do not reject L! since !Þ(' is not less than "Þ(&$ (c)!Þ" T !Þ#&