Assignmetn: Standard Deviation
1. True or False. Justify for fullcredit.
(a) The standard deviation of a data set cannot benegative.
(b) If P(A) = 0.4 , P(B) = 0.5, and A and B are disjoint, then P(A AND B) =0.2.
(c) The mean is always equal to the median for a normaldistribution.
(d) A 95% confidence interval is wider than a 98% confidence interval of the sameparameter.
(e) In a two-tailed test, the value of the test statistic is 1.5. If we know the test statistic follows a Student's t-distribution with P(T < 1.5) = 0.98, then we fail to reject the null hypothesis at 0.05 level of significance.
2. Identify which of these types of sampling is used: cluster, convenience, simple random, systematic, or stratified. Justify for fullcredit.
(a) A STAT 200 professor wants to estimate the study hours of his students. He teaches two sections, and plans on randomly selecting 10 students from the first section and 15 students from the secondsection.
(b) A STAT 200 student is interested in the number of credit cards owned by college students. She surveyed all of her classmates to collect sampledata.
(c) The quality control department of a semiconductor manufacturing company tests every 100thproduct from the assemblyline.
(d) On the day of the last presidential election, UMUC News Club organized an exit poll in which specific polling stations were randomly selected and all voters were surveyed as they left those pollingstations.
3. The frequency distribution below shows the distribution for checkout time (in minutes) in UMUC MiniMart between 3:00 and 4:00 PM on a Friday afternoon. (Show all work. Just the answer, without supporting work, will receive nocredit.)
|
Checkout Time (inminutes)
|
Frequency
|
Relative Frequency
|
|
1.0 -1.9
|
3
|
|
|
2.0 -2.9
|
12
|
|
|
3.0 -3.9
|
|
0.20
|
|
4.0 -4.9
|
3
|
|
|
5.0-5.9
|
|
|
|
Total
|
25
|
|
(a) Complete the frequency table with frequency and relative frequency. Express the relative frequency to two decimalplaces.
(b) What percentage of the checkout times was at least 4minutes?
(c) Does this distribution have positive skew or negative skew?Why?
4. A box contains 3 marbles, 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box, then replacing it in the box and drawing a second marble from the box. (Show all work. Just the answer, without supporting work, will receive nocredit.)
(a) List all outcomes in the samplespace.
(b) What is the probability that at least one marbleisred? (Express the answerinsimplest fractionform)
5. The five-number summary below shows the grade distribution of two STAT 200 quizzes for a sample of 500students.
|
Minimum
|
Q1
|
Median
|
Q3
|
Maximum
|
|
Quiz1
|
15
|
30
|
55
|
85
|
100
|
|
Quiz2
|
20
|
35
|
50
|
90
|
100
|
For each question, give your answer as one of the following: (i) Quiz 1; (ii) Quiz 2; (iii) Both quizzes have the same value requested; (iv) It is impossible to tell using only the given information. Then explain your answer in eachcase.
(a) Which quiz has less interquartile range in gradedistribution?
(b) Which quiz has the greater percentage of students with grades 90 andover?
(c) Which quiz has a greater percentage of students with grades less than50?
6. There are 1000 students in a high school. Among the 1000 students, 800 students have a laptop, and 300 students have a tablet. 250 students have both devices. Let L be the event that a randomly selected student has a laptop, and T be the event that a randomly selected student has a tablet. Show all work. Just the answer, without supporting work, will receive nocredit.
(a) Provide a written description of the event L ORT.
(b) What is the probability of event L ORT?
7. Consider rolling two fair dice. Let A be the event that the two dice land on differentnumbers, and B be the event that the first one lands on6.
(a) What is the probability that the first one lands on 6 given that the two dice land on different numbers? Show all work. Just the answer, without supporting work, will receive nocredit.
(b) Are event A and event B independent?Explain.
8. There are 8 books in the "Statistics is Fun" series. (Show all work. Just the answer, without supporting work, will receive nocredit).
(a) How many different ways can Mimi arrange the 8 books in her bookshelf?
(b) Mimi plans on bringing two of the eight books with her in a road trip. How many different ways can the two books beselected?
9. Assume random variable x follows a probability distribution shown in the table below. Determine the mean and standard deviation of x. Show all work. Just the answer, without supporting work, will receive nocredit.
|
x
|
-2
|
0
|
1
|
3
|
5
|
|
P(x)
|
0.1
|
0.2
|
0.3
|
0.1
|
0.3
|
10. Mimi just started her tennis class three weeks ago. On average, she is able to return 20% ofher opponent's serves. Assume her opponent serves 10times.
(a) Let X be the number of returns that Mimi gets. As we know, the distribution of X is a binomial probability distribution. What is the number of trials (n), probability of successes (p) and probability of failures (q),respectively?
(b) Find the probability that that she returns at least 1 of the 10 serves from her opponent. Show all work. Just the answer, without supporting work, will receive nocredit.
11. Assume the weights of men are normally distributed with a mean of 172 lb and a standard deviation of 30 lb. Show all work. Just the answer, without supporting work, will receive no credit.
(a) Find the 80th percentile for the distribution of men'sweights.
(b) What is the probability that a randomly selected man is greater than 185lb?
12. Assume the IQ scores of adults are normally distributed with a mean of 100 and a standard deviation of 15. Show all work. Just the answer, without supporting work, will receive nocredit.
(a) If a random sample of 25 adults is selected, what is the standard deviation of the samplemean?
(b) What is the probability that 25 randomly selected adults will have a mean IQ score that is between 95 and105?
13. A survey showed that 80% of the 1600 adult respondents believe in global warming. Construct a 95% confidence interval estimate of the proportion of adults believing in global warming. Show all work. Just the answer, without supporting work, will receive nocredit.
14. In a study designed to test the effectiveness of acupuncture for treating migraine, 100 patients were randomly selected and treated with acupuncture. After one-month treatment, the number of migraine attacks for the group had a mean of 2 and standard deviation of 1.5. Construct a 95% confidence interval estimate of the mean number of migraine attacks for people treated with acupuncture. Show all work. Just the answer, without supporting work, will receive nocredit.
15. Mimi is interested in testing the claim that more than 75% of the adults believe in global warming. She conducted a survey on a random sample of 100 adults. The survey showed that 80 adults in the sample believe in globalwarming.
Assume Mimi wants to use a 0.05 significance level to test theclaim.
(a) Identify the null hypothesis and the alternativehypothesis.
(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive nocredit.
(c) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive nocredit.
(d) Is there sufficient evidence to support the claim that more than 75% of the adults believe in global warming?Explain.
16. In a study of memory recall, 5 people were given 10 minutes to memorize a list of 20 words. Each was asked to list as many of the words as he or she could remember both 1 hour and 24 hours later. The result is shown in the followingtable.
|
Number of Words Recalled
|
|
Subject
|
1 hour later
|
24 hours later
|
|
1
|
14
|
12
|
|
2
|
18
|
15
|
|
3
|
11
|
9
|
|
4
|
13
|
12
|
|
5
|
12
|
12
|
Is there evidence to suggest that the mean number of words recalled after 1 hour exceedsthe mean recall after 24hours?
Assume we want to use a 0.10 significance level to test theclaim.
(a) Identify the null hypothesis and the alternativehypothesis.
(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive nocredit.
(c) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive nocredit.
(d) Is there sufficient evidence to support the claim that the mean number of wordsrecalledafter1 hour exceeds the mean recall after 24 hours? Justify yourconclusion.
17. The UMUC Daily News reported that the color distribution for plain M&M's was: 40% brown, 20% yellow, 20% orange, 10% green, and 10% tan. Each piece of candy in a random sample of 100 plain M&M's was classified according to color, and the results are listed below. Usea
0.05 significance level to test the claim that the published color distribution is correct. Show all work and justify youranswer.
|
Color
|
Brown
|
Yellow
|
Orange
|
Green
|
Tan
|
|
Number
|
42
|
21
|
12
|
7
|
18
|
(a) Identify the null hypothesis and the alternativehypothesis.
(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive nocredit.
(c) Determine the P-value. Show all work; writing the correct P-value, without supporting work, will receive nocredit.
(d) Is there sufficient evidence to support the claim that the published color distribution iscorrect? Justify youranswer.
18. A random sample of 4 professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions ofdollars).
(a) Find an equation of the least squares regression line. Show all work; writing the correct equation, without supporting work, will receive nocredit.
(b) Based on the equation from part (a), what is the predicted value of y if x = 3? Show all work and justify youranswer.
19. A farmer is interested in whether there is any variation in the weights of apples between two trees. Data collected from the two trees are asfollows:
Her null hypothesis and alternative hypothesisare:
(a) Determine the test statistic. Show all work; writing the correct teststatistic,without supporting work, will receive nocredit.
(b) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive nocredit.
(c) Is there sufficient evidence to justify the rejection of H0 at the significance level of 0.05? Explain.
20. A study of 5 different weight loss programs involved 250 subjects. Each program was followedby 50 subjects for 12 months. Weight change for each subject was recorded. Mimi wants to test the claim that the mean weight loss is the same for the 5programs.
(a) Complete the following ANOVA table with sum of squares, degrees of freedom, and mean square (Show allwork):
|
Source of Variation
|
Sum of Squares (SS)
|
Degrees of Freedom(df)
|
MeanSquare (MS)
|
|
Factor (Between)
|
42.36
|
|
|
|
Error (Within)
|
|
|
|
|
Total
|
1100.76
|
249
|
|
(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting work, will receive nocredit.
(c) Determine the P-value for this test. Show all work; writing the correct P-value, without supporting work, will receive nocredit.
(d) Is there sufficient evidence to support the claim that the mean weight loss is the same forthe 5 programs at the significance level of 0.01?Explain.