Questions:
1:
Prof. Hardtack gave four Friday quizzes last semester in his 10-student senior tax accounting class. Quiz 1:
60, 60, 60, 60, 71, 73, 74, 75, 88, 99 Quiz 2: 65, 65, 65, 65, 70, 74, 79, 79, 79, 79 Quiz 3: 66, 67, 70, 71, 72, 72, 74, 74, 95, 99 Quiz 4: 10, 49, 70, 80, 85, 88, 90, 93, 97, 98
(a) Find the mean, median, and mode for each quiz.
(b) Do these measures of center agree? Explain.
(c) For each data set, note strengths or weaknesses of each statistic of center.
(d) Are the data symmetric or skewed? If skewed, which direction? (e) Briefly describe and compare student performance on each quiz.
2:
In a sample of 100 Planter's Mixed Nuts, 19 were found to be almonds.
(a) Construct a 90 percent confidence interval for the true proportion of almonds.
(b) May normality be assumed? Explain.
(c) What sample size would be needed for 90 percent confidence and an error of ± 0.03?
(d) Why would a quality control manager at Planter's need to understand sampling?
3:
Consider the following Excel regression of perceived sound quality as a function of price for 27 stereo speakers.
(a) Is the coefficient of Price significantly different from zero at a = .05?
(b) What does the R2 tell you?
(c) Given these results, would you conclude that a higher prise implies higher sound quality?
4:
Suppose a pickup and delivery company states that their packages arrive within two days or less on average. You want to find out whether the actual average delivery time is longer than this. You conduct a hypothesis test.
(a) Set up the null and alternative hypotheses.
(b) Suppose you conclude wrongly that the company's statement
Answers:
1
- The mean, median and mode for each quiz is given in the following table:
|
|
Quiz 1
|
Quiz 2
|
Quiz 3
|
Quiz 4
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|
Mean
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72
|
72
|
76
|
76
|
|
Median
|
72
|
72
|
72
|
86.5
|
|
Mode
|
60
|
65
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72
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#N/A
|
- The median scores of the quizzes 1, 2 and 3 agree but the median score of quiz 4 is higher. Median scores indicate that half of the scores lie above the median score. Thus, it can be said that the performance of the students is better in Quiz 4. The mean score of quizzes 1 and 2 are same and quizzes 3 and 4 are same. Since the median of quiz 3 is same as quizzes 1 and 2, it can be said that there is an outlier, that is a score which is considerably higher than the other scores, hence affecting the mean. Maximum number of students scored 60 in Quiz 1, 65 in Quiz 2 and 72 in Quiz 3. No scores have been scored more than once in Quiz 4.
- For the datasets 1, 2 and 3, it can be seen that the scores are quite close to each other. Thus, mean gives a good measure of centre for these three datasets. On the other hand, the scores of quiz 4 are highly variable and thus, the mean gets affected. Hence, the median gives a better measure than the mean.
- The scores in quiz 1 and quiz 3 are positively skewed, the scores in quiz 2 are symmetrical and in quiz 4 the scores are negatively skewed.
- In quizzes 1 and 3, most of the students have scored less than the average score. In quiz 2, all the students have scored marks close to the average marks and in quiz 4, most of the students have scored more than the average score.
2
In a sample of 100 Planter’s Mixed Nuts, 19 nuts were almonds.
- Thus, the required 90 percent confidence interval for the true proportion of almonds has been expressed by the following table:
|
Confidence Interval
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Sample Size
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100
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Almonds
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19
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Proportion (p)
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(100/19) = 0.19
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Standard Error (SE)
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0.04
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|
Level of Significance (α)
|
0.10
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α/2
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0.05
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Z Value (z)
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1.64
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Lower Confidence Limit
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(p – z * SE) = 0.13
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Upper Confidence Limit
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(p + z * SE) = 0.25
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- Normality has to be assumed. Without the assumption of normality, no parametric tests can be performed.
- The required sample size necessary for 90 percent confidence and a margin of error of ±03 has been calculated and given in the following table:
|
Estimation of Sample Size
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Proportion (p)
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0.19
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|
p (1-p)
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0.1539
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|
Margin of Error (E)
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0.03
|
|
Level of Significance (α)
|
0.1
|
|
α/2
|
0.05
|
|
Z Value (z)
|
1.64
|
|
Sample Size
|
462.65
|
- It is important for the quality control manager at planters to understand the concept of sampling. The company manufactures different types of nuts. It is not possible to find out the total number of each type of nuts that the company manufactures. Thus, the proportional test on the samples has to be conducted to find out the range within which the proportion of each type of nuts are produced by the company.
3
- It can be seen from the results of the regression analysis provided in the question that the p-value of the coefficient of price is 0.6019 which is more than the level of significance (α) which is 0.05. This, it can be said that the variable price is insignificant and the coefficient of price is not significantly different from zero.
- The value of R Square, which is found to be 0.01104 indicates that 1.104 percent of the variability in the perceived sound quality of the stereo speakers can be explained by the price of the speakers.
- Given these results, it can be concluded that the sound quality of the speakers does not depend on the price of the speakers. There is no such impact of price on the sound quality of the speakers.
4
- Null Hypothesis:The delivery time of the package is less than or equal to two days on an average.
Alternative Hypothesis: The delivery time of the package more than two days on an average.
- If it is concluded wrongly that the claim of the company that the average delivery time is within two days, then the true null hypothesis will be rejected. This is Type I error. With this type of error, the company will be wrongly accused about the delivery service and the orders received by the company might decrease due to the longer delivery time as found from the test.
- Now, the true fact was that the delivery time is longer than two days and by testing, it has been concluded that the delivery time is within 2 days. This type of error is termed as Type II error as the false null hypothesis is accepted. For the company this will be good fact as their delivery time will be termed as less when that is not actually the case. This will be beneficial for the company.
- Thus, from the company’s standpoint, Type I error is worse as that error will be affecting the business of the company.
- From the standpoint of the consumer, Type II error will be worse as this will be affecting the service received by the consumers.
