Use R to complete the following questions. You should include your R code, output and plots in your answer.
1. Two methods of generating a standard normal random variable are:
a. Take the sum of 5 uniform (0,1) random numbers and scale to have mean 0 and standard deviation 1. (Use the properties of the uniform distribution to determine the required transformation).
b. Generate a standard uniform and then apply inverse cdf function to obtain a normal random variate (Hint: use qnorm).
For each method generate 10,000 random numbers and check the distribution using
a. Normal probability plot
b. Mean and standard deviation
c. The proportion of the data lying within the theoretical 2.5 and 97.5 percentiles and the 0.5 and 99.5 percentiles. (Hint: The ifelse function will be useful)
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These problems may be solved using Minitab. Copy and paste the appropriate Minitab output into a word-processed file. Add your explanations of the output near the Minitab output. DO NOT SIMPLY ATTACH PAGES OF OUTPUT AS AN APPENDIX.
Each problem should be able to fit on one or two pages, and each problem should include the following:
Design 1 |
Design 2 |
Design 3 |
Design 4 |
Design 5 |
12.2 |
12.2 |
10.0 |
10.2 |
11.0 |
12.4 |
13.4 |
11.2 |
7.9 |
12.5 |
11.9 |
12.4 |
8.9 |
9.1 |
11.7 |
11.7 |
11.0 |
11.2 |
11.2 |
10.8 |
11.7 |
12.4 |
10.2 |
10.1 |
10.0 |
12.0 |
13.1 |
10.6 |
6.6 |
9.8 |
11.8 |
11.5 |
10.4 |
8.1 |
10.3 |
11.5 |
11.6 |
9.2 |
10.0 |
9.3 |
13.9 |
13.3 |
10.8 |
8.7 |
11.1 |
13.2 |
12.7 |
11.5 |
8.4 |
12.9 |
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2. Lactation promotes a temporary loss of bone mass to provide adequate
amounts of calcium for milk production. Consider the data on total
body bone mineral content for a sample both during lactation (L) and
in the post weaning period (P).
1 . 2 3 4 5 6 7 8 9 10
L 1928 2549 2825 1924 1628 2175 2114 2621 1843 2541
P 2126 2885 2895 1942 1750 2184 2164 2626 2006 2627
Does the data suggest that true average total body bone mineral con-
tent during post weaning exceeds that during lactation by more than
25g? Use results from an output you obtained from the R software to
state and test the appropriate hypotheses with 95% confidence level(show
all details of R, either print our the R code and result or hand-write
R code and result ). State any assumptions required for the test to be
valid.
5. The following table summarizes the skin colours and position of 368
NBA players in 2014. Suppose that an NBA player is randomly selected
from that years player pool.
2
Guard positionForward Center Total
white 26 30 28 84
skin colour black 128 122 34 284
Total 154 152 62 368
(a) Find out where the variable "skin colour" is independent with the
variable "position" with
alpha = 0.05
(b) We only concern about the variable "position". The media claims
that the proportion of "Guard" is the same as the proportion of "For-
ward", which is twice as the proportion of "Center". Conduct a test to
find out whether the statement is valid with 90% confidence interval.
(hint: to test H0: P1= 0.4, P2 = 0.4, P3= 0.2)
6. Complete the following ANOVA table ( find the value of ?1, ?2, ?3, ?4,
?5) [5] and give the null hypothesis and the alternative hypothesis, [2]
give your conclusion based on the ANOVA table [2].
ANOVA table
Df SumSq Mean Sq F value P value
brands 3 39.757 ?1 ?5 5.399e-07
Error 36 ?2 ?3
Total ?4 . 68.128
7. Refer to the
Bulletin of Marine Science (April 2010)
study of teams
of shermen shing for the red apiny lobster in Baja Valifornia Sur,
Mexico. Two variables measured for each of 8 teams from the Punta
Abreojos shing cooperative were y=total catch of lobsters (in kilo-
grams) during the season and x=average percentage of traps allocated
per day to exploring areas of unknown catch (called search frequency)
total catch search frequency
2785 35
6535 21
6695 26
4891 29
4937 23
5727 17
7019 21
5735 20
(a) Graph the data in a scatterplot (using R). What type of trend, if any,
could be observed?
(b) Add the regression line to the plot (using R, either hand-write the R
code and plot the graph or print your R code and graph).
(c)Give the null and alternative hypothesis for testing whether total catch
is negatively linearly related to search frequency. Find the p-vale of the test
and give the appropriate conclusion of the test using alpha = 0.05
(d) what's the coefficient of correlation between total catch and search
frequency?
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What are the advantage and disadvantage of assuming quadratic utility functions in mean variance analysis?
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Construct a scatter plot. Find the equation of the regression line. Predict the value of y for each of the x-values. Use this resource: Regression Give an example of two variables that have a positive linear correlation.
Give an example of two variables that have a negative linear correlation.
Give an example of two variables that have no correlation.
Height and Weight: The height (in inches) and weights (in pounds) of eleven football players are shown in this table.
Height, x 62 63 66 68 70 72 73 74 74 75 75 Weight, y 195 190 250 220 250 255 260 275 280 295 300
x = 65 inches x = 69 inches x = 71 inches
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A random sample of 130 observations produced a mean of ?⎯⎯⎯=36.1x¯=36.1 from a population with a normal distribution and a standard deviation σ=4.87.
(a) Find a 95% confidence interval for μ
≤ μ ≤
(b) Find a 99% confidence interval for μ
≤ μ ≤
(c) Find a 90% confidence interval for μ
≤ μ ≤
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Given are five observations for two variables, x and y. x i 1 2 3 4 5 y i 4 7 6 11 13 Round your answers to two decimal places. a. Using the following equation: Estimate the standard deviation of ŷ* when x = 3. b. Using the following expression: Develop a 95% confidence interval for the expected value of y when x = 3. to c. Using the following equation: Estimate the standard deviation of an individual value of y when x = 3. d. Using the following expression: Develop a 95% prediction interval for y when x = 3. If your answer is negative, enter minus (-) sign. to
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The average number of words in a romance novel is 64,290 and the
standard deviation is 17,422. Assume the distribution is normal.
Let X be the number of words in a randomly selected romance novel.
Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(___,____)
b. Find the proportion of all novels that are between 57,321 and
71,259 words. _____
c. The 95th percentile for novels is ____ words. (Round to the
nearest word)
d. The middle 50% of romance novels have from ____words to_____
words. (Round to the nearest word)
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find the sample size needed to give with 99% confidence a margin of error of plus or minus 5% when estimating proportion within plus minus 4% within plus minus 1%
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The mean output of a certain type of amplifier is 496 watts with a variance of 144. If 40 amplifiers are sampled, what is the probability that the mean of the sample would differ from the population mean by less than 1.2 watts? Round your answer to four decimal places.
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The College Board wanted to test whether students graduating from private colleges and students graduating from public universities had different amounts of student loan debt. A sample of students from 146 private colleges across the country yielded an average loan debt of $29,972 with a standard deviation of $3,200. A sample of students from 225 public universities yielded an average loan debt of $28,762 with a standard deviation of $5,600. Conduct the test at the α=0.02α=0.02 level of significance.
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What is meant by an absolute effect in epidemiologic research?
Present at least one relevant example.
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The mean cost of domestic airfares in the United States rose to an all-time high of $380 per ticket. Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of $100. Use Table 1 in Appendix B.
a. What is the probability that a domestic
airfare is $545 or more (to 4 decimals)?
b. What is the probability that a domestic
airfare is $255 or less (to 4 decimals)?
c. What if the probability that a domestic
airfare is between $320 and $490 (to 4 decimals)?
d. What is the cost for the 5% highest domestic airfares? (rounded to nearest dollar)
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19. Assume that females have pulse rates that are normally distributed with a mean of mu equals 74.0 beats per minute and a standard deviation of sigma equals 12.5 beats per minute. Complete parts (a) through (c) below.
a. If 1 adult female is randomly selected, find the probability that her pulse rate is between 70 beats per minute and 78 beats per minute.
The probability is _____
(Round to four decimal places as needed.)
b. If 4 adult females are randomly selected, find the probability that they have pulse rates with a mean between 70 beats per minute and 78 beats per minute.
The probability is____
(Round to four decimal places as needed.)
c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?
A.Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.
B.Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size.
C.Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size.
D.Since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size.
20. An elevator has a placard stating that the maximum capacity is 2310 lb -15 passengers. So, 15 adult male passengers can have a mean weight of up to 2310 divided by 15 equals 154 pounds. If the elevator is loaded with 15 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 154 lb. (Assume that weights of males are normally distributed with a mean of 163 lb and a standard deviation of 31 lb.) Does this elevator appear to be safe?
The probability the elevator is overloaded is_____
(Round to four decimal places as needed.)
Does this elevator appear to be safe?
A.No, there is a good chance that 15 randomly selected people will exceed the elevator capacity.
B.Yes, there is a good chance that 15 randomly selected people will not exceed the elevator capacity.
C.Yes, 15 randomly selected people will always be under the weight limit.
D.No, 15 randomly selected people will never be under the weight limit.
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The world is full of misleading messages. Many of them are comming from the fact that people do not know how to interpret data. Find an example of a misleading use of statistics in a newspaper, magazine, corporate annual report, or other source. Then explain why your example is misleading.
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