MA416 - Analysis of Variance - Mini-Exam 1
Directions (Please Read) : The time permitted for this mini-exam is 35 minutes. You must show all work to receive full, if any, credit (where applicable). A basic scientific or TI calculator may be used, in addition to a 3” x 5” note card (handwritten, back and front); no notes, books, search engine, cell phone, any other person, device, or resource aside from yourself can be used to get assistance on any portion of this exam. Failure to adhere to any of these (or university) policies will result in a zero for this assessment. Please try to adhere to the recommended time limits on each question block.
[3 mins] For the following (1) – (4),circle either “True” or “False” regarding the validity of the entire statement provided.
1. True
2. True
3. True
4. True
False If you have evidence to reject H0 : σA(2) = σB(2), then we have evidence to believe that the populations for which
the associated sample variables came from are homoscedastic.
False Suppose you have a sample such that the arithmetic mean, median, mid-range, and mode are all equal to
one another. Then that sample must be normally distributed.
False If the arithmetic mean of a sample data set is greater than the median of the same sample data set, then
we have evidence to believe that the population the sample was collected from is skewed left.
False If a sample data set contains only positive numbers, then taking the natural logarithm of all of the sample
values will always reduce the skewness of that data set.
[12 mins] For the following (5) – (8), you are interested in seeing if there appears to be a dependency between favorite color and favorite music genre. You sample a series of individuals and obtain the following results.
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Music = Pop
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Music = Country
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Color = Red
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32
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13
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Color = Green
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42
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25
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Color = Blue
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38
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27
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5. How many people did you survey?
6. What would be the associated degrees of freedom for this test for independence? Show work to support your answer.
7. If favorite color and music were truly independent, what would be your expected frequency of sample respondents whose favorite color is blue and favorite music type is country? Show work to justify your answer.
8. Suppose you obtain achi-square test statistic of 76.52, and suppose for some an alpha of 5%. Below are the potential critical values. What would your conclusion be? Be very clear in your answer and reasoning.
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Test Type/Location
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Two-Tail Left
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Two-Tail Right
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One-Tail Left Tail
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One-Tail Right Tail
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X 2 Critical Value
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19.8063
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51.9660
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21.6643
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48.6024
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[15 mins] For the following (9) - (11), suppose you collect a simple random sample from a population whose summary statistics are below.
n = 23 ̅(x) = 76.5 sX = 5.7 Q0 = 45.2 Q1 = 62.3 Q2 = 74.5 Q3 = 77.9 Q4 = 98.7
Use the critical values as they are mentioned below for some alpha value.
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Test Type/Location
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Two-Tail Left
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Two-Tail Right
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One-Tail Left
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One-Tail Right
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Z Critical Value
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-1.9599
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1.9599
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-1.6449
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1.6449
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T Critical Value, v = 22
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-2.0739
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2.0739
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-1.7171
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1.7171
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T Critical Value, v = 21
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-2.0796
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2.0796
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-1.7207
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1.7207
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X Critical Value, v = 22
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3.3140
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6.0647
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3.5125
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5.8245
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X 2 Critical Value, v = 22
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10.9823
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36.7807
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12.3380
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33.9244
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F Critical Value, v = (22,23)
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0.4266
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2.3254
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0.4908
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2.0246
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F Critical Value, v = (23,22)
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0.4300
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2.3442
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0.4939
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2.0377
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9. Someone claims that the population mean is equal to 79. Construct a confidence interval for the true population mean, and then conclude your results. Be clear and organized in your answer. Show all work to support your answer. Round to 2 decimal places if needed.
10. Someone claims that the population standard deviation is equal to 8.3. Construct a confidence interval for the true population standard deviation, and then conclude your results. Be clear and organized in your answer. Show all work to support your answer. Round to 2 decimal places if needed.
11. Suppose that a sample that was drawn from another population was sent to you. The sample size was 24, and had a sample mean of 73.5 and a sample standard deviation of 5.3. You plan to merge these samples together if they are homoscedastic to one another. Construct a test statistic for this criterion, and then conclude your results. Be clear and organized in your answer. Show all work to support your answer. Round to 2 decimal places if needed.
Solution Guide
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1.
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F
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2.
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F
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3.
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F
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4.
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F
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5.
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177
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6.
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2
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7.
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23.87
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8. Since (Xs(2)tat = 76.5) > (XC(2)Tit = 48.6), we have evidence to believe that the variables are dependent on one another.
9. (CI)μ = (74.04, 78.96), μ0 = 79, we have evidence to believe that the claim that the mean is 79 is incorrect.
10. (CI)σ = (4.41, 8.07), σ0 = 8.3, we have evidence to believe that the claim that the standard deviation is 8.3 is incorrect. 11. Fstat = 1.16, FCTit = 2.0246 (make sure you know why it’sthis one). Since the test statistic is less than the critical value, we have evidence to believe the populations are not heteroscedastic, and hence we should merge the samples together.