Question

Question 1 – High Technology Stocks [10 marks]

A random sample of high technology stocks was followed over a month to determine whether there has been an overall increase in the price of hi-tech stock shares (from the cost price per share a month ago to the current market price per share). Refer to Appendix A for the data and analysis.

1. [1 mark] Without reference to the data distributions but considering the manner in which the data were collected, identify two appropriate tests for determining whether there has been an overall increase in prices.

2. [2 marks] Now look at the boxplots of the data. Which of the two tests above is the most appropriate? Explain briefly with specific references to the appropriate boxplot(s).

3. [4 marks] Ignoring your answer in b), perform the most appropriate parametric test to determine if there has been an overall increase in prices. Use a 10% significance level.

4. [3 marks] Ignoring your answer in b), perform the most appropriate non-parametric test.

Appendix A

MarketPrice	Costprice	Diff
63.4	88.96	-25.56
8.92	8.43	0.49
6.12	6.37	-0.25
15.771	11.5	4.271
47.415	44.94	2.475
3.22	3.55	-0.33
93.12	94.44	-1.32
41.624	28.17	13.454
45.95	38.79	7.16
5.41	5.23	0.18
6.05	4.61	1.44
3.68	4.01	-0.33
4.04	6.55	-2.51
23.1	20.44	2.66

Boxplot of MarketPrice, Costprice

Two-Sample T-Test and CI: MarketPrice, Costprice

Two-sample T for MarketPrice vs Costprice

              N Mean StDev SE Mean

MarketPrice 14 26.3   27.9      7.5

Costprice    14 26.1   30.8      8.2

Difference = mu (MarketPrice) - mu (Costprice)

Estimate for difference: 0.130714

90% lower bound for difference: -14.501981

T-Test of difference = 0 (vs >): T-Value = XXXX P-Value = XXXX DF = 25

Paired T-Test and CI: MarketPrice, Costprice

Paired T for MarketPrice - Costprice

              N      Mean     StDev   SE Mean

MarketPrice 14   26.2729   27.9076    7.4586

Costprice    14   26.1421   30.8404    8.2425

Difference   14 0.130714 8.446491 2.257420

90% lower bound for mean difference: -2.917189

T-Test of mean difference = 0 (vs > 0): T-Value = XXXX P-Value = XXXX

Mann-Whitney Test and CI: MarketPrice, Costprice

              N Median

MarketPrice 14   12.35

Costprice    14    9.97

Point estimate for η1 - η2 is 0.08

90.6 Percent CI for η1 - η2 is (-7.46,12.23)

W = 204.0 (test statistic)

Test of η₁ = η₂ vs η₁ > η₂ is significant at 0.4908 (p-value)

Note: η in the output above denotes “true median”.

Wilcoxon Signed Rank Test: Diff

Test of median = 0.000000 versus median > 0.000000

             N

           for   Wilcoxon         Estimated

       N Test Statistic p-value     Median

Diff 14    14       67.0 0.190     0.8805

Answer 1

1. [1 mark] Without reference to the data distributions but considering the manner in which the data were collected, identify two appropriate tests for determining whether there has been an overall increase in prices.

Paired t-test and Wilcoxon Signed Rank Test.

1. [2 marks] Now look at the boxplots of the data. Which of the two tests above is the most appropriate? Explain briefly with specific references to the appropriate boxplot(s).

The boxplot is:

Since the data has an outlier, it is appropriate to use the non-parametric test.

1. [4 marks] Ignoring your answer in b), perform the most appropriate parametric test to determine if there has been an overall increase in prices. Use a 10% significance level.

The hypothesis being tested is:

H0: µd = 0

Ha: µd > 0

26.272857	mean MarketPrice
26.142143	mean Costprice
0.130714	mean difference (MarketPrice - Costprice)
8.446491	std. dev.
2.257420	std. error
14	n
13	df
0.058	t
.4774	p-value (one-tailed, upper)

The p-value is 0.4774.

Since the p-value (0.4774) is greater than the significance level (0.10), we cannot reject the null hypothesis.

Therefore, we cannot conclude that there has been an overall increase in prices.

1. [3 marks] Ignoring your answer in b), perform the most appropriate non-parametric test.

The hypothesis being tested is:

H0: η1 = η2

Ha: η1 > η2

variables:	MarketPrice - Costprice
67	sum of positive ranks
38	sum of negative ranks
14	n
52.500	expected value
15.930	standard deviation
0.910	z
.1813	p-value (one-tailed, upper)

The p-value is 0.1813.

Since the p-value (0.1813) is greater than the significance level (0.10), we cannot reject the null hypothesis.

Therefore, we cannot conclude that there has been an overall increase in prices.

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