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STA101 Statistics For Business
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STA101 Statistics For Business
1 Download4 Pages / 984 Words
Course Code: STA101
University: Charles Darwin University
MyAssignmentHelp.com is not sponsored or endorsed by this college or university
Country: Australia
Question:
Two contractors, Bill and Ben have put in the same bid for a landscaping job. You want to decide between them based on their past performance at completing jobs. The following tables are probability distributions showing the amount of time it takes Bill & Ben to finish similar jobs.
Please answer following 6 questions:
(a) How long on average would it take (i) Bill, (ii) Ben to finish the job?
(b) What is the standard deviation of the time it would take each of Bill and Ben to finish the job? Interpret the results.
(c) Who would you choose to do the job?
The following dataset contains ages of 10 randomly selected students in a school. Complete (a) to (e) below. (6 marks)
(a) Compute the mean and standard deviation.
(b) Find the median, and compare the mean and median to determine the skewness of the dataset.
(c) Calculate the coefficient of variation.
(d) Locate the first quartile, third quartile
(e) Locate the 80th percentile.
(e) Find out the range and inter-quartile range for the dataset. Suppose the last observation (42) was incorrectly entered. If the correct age was 80, how would the correction affect the range and the inter-quartile range?
The clean air legislation in Australia requires that vehicle exhaust emissions do not exceed certain limits for various types of pollutants. In Queensland, the Department of Environment has recently received complaints that some car repair workshops certified vehicles that did not meet the emission standard. The regulator decides to check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shop’s license if they find significant evidence that the shop is issuing certificates for cars that do not meet the standard.
(a) In this context, what is a type I error?
(b) In this context, what is a type II error?
(c) Which type of error would the shop owner consider more serious?
(d) Which type of error might the environmentalists consider more serious?
Why does σ pose a problem for sample size calculation for a mean? How can σ be approximated when it is unknown?
Why is it better to say “fail to reject H0” instead of “accept H0”? (6 marks)
Answer:
Answer 1.
Bill
Total
Weeks (X)
3
4
5
6
4.5
P(X=x)
0.2
0.3
0.3
0.2
X*P(X=x)
0.6
1.2
1.5
1.2
4.5
(x-μ)2
2.25
0.25
0.25
2.25
1.024695
(Standard deviation)
(x-μ)2*P(X=x)
0.45
0.075
0.075
0.45
Ben
Total
Weeks (X)
2
4
6
7
4.75
P(X=x)
0.2
0.4
0.3
0.1
X*P(X=x)
0.4
1.6
1.8
0.7
4.5
(x-μ)2
7.5625
0.5625
1.5625
5.0625
7.885905
(Standard deviation)
(x-μ)2*P(X=x)
15.125
2.25
9.375
35.4375
a)
a. i) Bill would take 4.5 weeks to complete the job on an average.
a. ii) Ben would take on an average 4.5 weeks to complete the job.
b)
The standard deviation of the time to take each of Bill and Ben to finish the job are 1.024695 and 7.885905 respectively.
The scatter-ness of the time taking probability is higher for Ben than Bill.
c)
Both of Bill and Ben need equal average of time to finish the job; however, the scatter-ness of the time taking probability is lower for Ben. Hence, Bill should be chosen to finish the job.
Answer 2.
Age
Revised age
9
9
11
11
12
12
13
13
13
13
15
15
16
16
21
21
28
28
42
80
Mean
18
Standard deviation
10.07748
Median
14
Coefficient of variation
55.986%
First quartile
12.25
12.25
Third quartile
19.75
19.75
80% percentile
22.4
Range
33
71
Inter-quartile range
7.5
7.5
a) Calculated mean (μ) = 18.
Calculated standard deviation (σ) = 10.07748.
b) Calculated median () = 14.
The undertaken data sample includes outlier such as 42. Also, the data set is skewed. It is the reason that median is the better measure than mean. Mean is highly influenced by outliers and skew data; however, median is not.
c) Calculated co-efficient of variation () = 55.986%.
d) The first quartile (Q1) of the data set = 12.25.
The third quartile (Q3) of the data set = 19.75.
e) The 80% of the dataset = 22.4.
f) The range of the dataset = 33.
The inter-quartile range of the dataset = 7.5.
When the last observation (42) was incorrectly entered and the age is corrected to be 80, then the range (= (maximum-minimum)) becomes 71 from 33. However, the inter-quartile range did not change from 7.5. As, range depends on outlier values that may include maximum and minimum values, the range has changed after correction. However, the inter-quartile range is dependent upon two location measures (IQR = Q3 – Q1) (Rodriguez, Reise and Haviland 2016). Both the quarters are same for previous age data and revised age data. Therefore, the correction has not affected the inter-quartile range.
Answer 3.
P (0
P (Z<-2 or >2) = 0.5* P (|Z|>2) = = 0.02275.
P (Z>0.6 and <2.5) = (1- P (Z<0.6) - P (Z>2.5)) = (1 – 0.7257 – 0.00621) = (1- (0.7257+0.00621)) = (1- 0.73191) = 0.26809.
The z-value so that the left-tail is 0.2 = P (Z<0.2) = 0.5793.
The symmetric Z-values that contain 95% of all observations = 1.96.
Answer 4.
a) In this context, if a car does not meet the emission-standard and although the license of the standard is not removed, then type-I error would occur (Sullivan and Feinn 2012).
b) In this context, if a car meets the standard of emission; however, the license of the standard is removed, then type-II error would occur.
c) Type-II is considered more serious as per shop-owner. It is very unpleasant to a shop-owner while a car manufactured by him/her meets the standard of emission, but the license gets cancelled. While it is not at all demotivating for him if a car without meeting the emission-standard does not get a block as per type-I error.
d) Type-I is considered more serious as per environmentalist. It is unjustified from the end of an environmentalist while a car does not meet the emission-standard, but the license is not cancelled. Type-II is not so much serious from a point of view of an environmentalist where meeting the standard of emission, the license of a car gets cancelled.
Answer 5.
The standard deviation (σ) of unknown population should not be too small. Otherwise, the sample size would be very small and would fail to validate the reflection of large population. That is why sometimes the standard deviation (σ) pose a problem for sample size calculation for a mean. The lack of effectiveness of an intervention of legitimate standard deviation would make the data set reliable and valid.
In the case, where the population is unknown, the sample size could be derived by-
Compute the required minimum sample size for accuracy to estimate accuracy.
Consider the standard normal deviation set at 5% level of significance (95% confidence level).
Pick up the percentage choice or response (proportion) at that level of significance.
The calculated sample size (n) = = (Burmeister and Aitken 2012)
Here, z = standard normal deviation set at 5% level of significance, p = proportion or the percentage of choice, c = confidence interval, s2 = population standard deviation, d = acceptable standard error (SE) of mean.
Answer 6.
Most of the statisticians prefer to say 'fail to reject' null hypothesis rather than accepting the null hypothesis. One statistician should always try to reject the null hypothesis with preferable level of significance such as 1%, 5% or 10%. He/she must find enough evidences to reject the null hypothesis. The probability of null hypothesis that is not rejected for a certain level of significance could be rejected at lower level of significance and higher critical region (Johnson 2013). Hence, it is justified to say 'fail to reject' null hypothesis rather than saying 'accepting' the null hypothesis.
References:
Burmeister, E. and Aitken, L.M., 2012. Sample size: How many is enough?. Australian Critical Care, 25(4), pp.271-274.
Johnson, V.E., 2013. Revised standards for statistical evidence. Proceedings of the National Academy of Sciences, 110(48), pp.19313-19317.
Rodriguez, A., Reise, S.P. and Haviland, M.G., 2016. Evaluating bifactor models: Calculating and interpreting statistical indices. Psychological methods, 21(2), p.137.
Sullivan, G.M. and Feinn, R., 2012. Using effect size—or why the P value is not enough. Journal of graduate medical education, 4(3), pp.279-282.
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