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## Stem-and-leaf displays and frequency plots

(a)List all the quarterly opening price values in two tables, one for MQG and the other for PPT. Then construct a stem-and-leaf display with one stem value in the middle, and MQG leaves on the right side and PPT leaves on the left side. (Must use EXCEL or similar for the plot.)

(b)Construct a relative frequency histogram for MQG and a frequency polygon for PPT on the same graph with equal class widths, the first class being “\$0 to less than \$10”. Use two different colours for MQG and PPT. Graph must be done in EXCEL or similar software.

(c)Draw a bar chart of market capitals (or total assets) in 2017 (in million Australian dollars) of 6 companies listed in ASX that trade in financials with over AUD500 million in market capital.  Graphing must be done in EXCEL or with similar software.

If one wishes to invest in MQG or PPT, what is the market recommendation (for example, from Morningstar, Fatprophets, InvestSmart, etc.)? If you cannot find the information, what would be your recommendation based on your research of these two companies (trend, P/E ratio, dividend yield and Beta)?

...
(a)ompute the mean, median, first quartile, and third quartile of average price-earnings ratio for each company using the exact position, (n+1)f, where n is the number of observations and f the relevant fraction for the quartile.

(c)Draw a box and whisker plot for the average price-earnings ratio of each company and put them side by side on one graph with the same scale so that the P/Es can be compared. (This graph must be done in EXCEL or similar software and cannot be hand-drawn.)

Write a paragraph on price-earnings ratio values related to it being indicative of a share being a good investment or a poor investment.

(a)Based on the table above, what is the probability that an Australian selected at random will die from neoplasms?

(b)Based on the table above, what is the probability that an Australian selected at random will die from a disease of the circulatory system and is a female?

(c)Based on the table, which disease has the highest proportion of male deaths compared to female deaths, and also, which disease has the highest proportion of female deaths compared to male deaths?

Visit the ABS website (Data Cube 1) and determine the probability of an Australian selected at random of dying from diabetes mellitus.

Assuming that the weekly total amount of rainfall (in mm) from the data provided in part (a) has a normal distribution, compute the mean and standard deviation of weekly totals.

(i) What is the probability that in a given week there will be between 5mm and 15mm of rainfall?

(ii) What is the amount of rainfall if only 10% of the weeks have that amount of rainfall or higher?

(a) Test for normality of all the variables separately using normal probability plot, except the first and last column variables.

(b)  Construct a 90% confidence interval for each of the variables in part (a) which can be considered as normally distributed separating the data between float glass and non-float glass. In the last column, the numbers 1 and 3 represent float glass, all other numbers indicate non-float glass. In other words, there will be two confidence interval constructions for each normally distributed variable – one for the float glass observations and the other for the non-float glass observations. After all the confidence intervals have been constructed, identify the variable(s) which are significantly different in composition between float and non-float type of glasses (i.e., identify the variables where the confidence intervals do not overlap).

## Stem-and-leaf displays and frequency plots

1. Quarterly opening prices
 Year Quarter MQG PPT Year Quarter MQG PPT 2007 1 79.65 78.45 2013 1 37.86 38.84 2 85.56 80.91 2 38.54 41.20 3 81.13 77.50 3 43.16 39.90 4 82.66 74.26 4 50.11 46.00 2008 1 63.92 58.30 2014 1 54.09 47.04 2 62.37 54.84 2 57.70 46.90 3 50.58 40.76 3 58.52 48.82 4 28.91 34.93 4 61.17 46.45 2009 1 25.56 30.72 2015 1 62.15 49.18 2 32.93 32.18 2 77.97 54.16 3 43.29 33.51 3 82.15 44.85 4 49.17 37.62 4 85.70 44.91 2010 1 49.33 35.05 2016 1 71.58 41.31 2 49.46 34.55 2 63.50 42.75 3 36.58 29.03 3 74.39 45.50 4 35.60 37.85 4 79.80 45.24 2011 1 39.95 31.29 2017 1 84.60 46.80 2 34.58 28.89 2 93.00 52.78 3 27.11 23.58 3 85.83 50.48 4 24.50 22.75 4 98.35 48.48 2012 1 25.08 20.25 2 28.73 25.50 3 24.57 23.91 4 31.37 27.94
 MQG and PPT Stem-and-Leaf Plot                      Stem &  Leaf                98753320 . 2 .  4455788             98775443210 . 3 .  12456789        9887666655442110 . 4 .  33999                    4420 . 5 .  00478                         . 6 .  12233                     874 . 7 .  14799                       1 . 8 .  1224555                         . 9 .  38  Each leaf: 1 case(s)

(Fletcher, 2009)

1. Histogram and Frequency polygon for MQG and PPT

Figure 1: Histogram and Frequency polygon of MQG and PPT

1. Market capitals for 6 companies listed in ASX with over AUD 500 million

Figure 2: Market capitals

1. Invest in MQG or PPT

Based on the series of previous share prices from 2007 to March 2018, I would advise on investing with PPT because the prices are down and there is a possibility of a rise in a few moments. This is in comparison to MQG whose prices have been rising since 2010 and just as any other stock price, there are very high chance of the prices losing value. Therefore, it would be more advisable to invest in a stock whose price has a higher chance of increasing than decreasing. In addition, PPT has a higher dividend yield of 6.65% compared to MQG which is 4.66%. Further, PPT has a lower Price-earning Ration compared to MQG, which make PPT stock better to invest in because their future growth has higher chances of growth(Aspara, 2009; Lynott, 2005).

1. Mean, median, 1stand 3rd Quartiles
 TLS SPK TPM CNU VOC Mean 13.37 13.33 18.65556 11.76667 21.34 Median 13.75 12.85 15.8 10.95 22.25 First Quartile 10.85 11.5 14.2 6.7 12.625 Third Quartile 15.475 15.55 25.3 17.475 28.9
1. Standard Deviation, Mean Absolute Distance and Range statistics
 TLS SPK TPM CNU VOC Standard Deviation 2.322951 2.148046 6.225951 4.764335 10.46501 Mean Absolute Distance 2.056 1.936 5.617284 4.533333 8.56 Range 6.8 6.2 19.2 11.3 36.6

(Cohen, Manion, & Morrison, 2011)

1. Box and Whisker Plots

Figure 3: Box and Whisker Plots

1. A discussion on the price-earnings ratio values

Higher values of price-earnings ratio show that the stock is overpriced while stocks with small values are seen to be better because they have higher growth potentials. Therefore, CNU has the highest growth potential compared to the others because the median measure small compared to the others. However, SPK has the least margin for the 3rd and 1st quartile, hence lower deviation from the median. It is very risky to invest in VOC stock because it is hard to estimate its ideal price-earnings ratio because of the higher variation. Therefore, it will be safer to invest in CNU, SPK and TLS stock as opposed to TPM and VOC(Titman, Wei, & Xie, 2004).

1. The probability that an Australian will die from neoplasms.
1. The probability of a Female Australian to die from disease of the circulatory system
1. Proportions of deaths

The disease with the highest proportion of male deaths compared to female deaths is Neoplasms with a difference of 6.3%. Diseases of circulatory system such as heart disease have the highest difference in proportion between female deaths and male deaths with a difference of 3.4%.

1. The probability of dying from diabetes mellitu

(McCluskey & Lalkhen, 2007)

1. Probability of rainfall
2. The probability that there will be no rainfall in any given day
1. The probability of 2 or more days of rainfall in a week
1. Assuming that the weekly rainfall follows a normal distribution

The mean weekly rainfall is 9.9132mm and a standard deviation of 12.297mm

1. The probability of having rainfall between 5mm to 15mm
1. Amount of rainfall if only 10% of the weeks have that amount of rainfall or higher

(Tsokos, Wooten, Tsokos, & Wooten, 2016)

The amount of rainfall will be

1. Normality Test

Refractive Index

Figure 4: Refractive index for refractive index

Sodium

Figure 5: Probability plot for Sodium

Magnesium

Figure 6: Probability plot of Magnesium

Aluminium

Figure 7: Probability plot of Aluminium

Silicon

Figure 8: Probability plot of Silicon

Potassium

Figure 9: Probability plot of Potassium

Calcium

Figure 10: Probability plot of Calcium

Barium

Figure 11: Probability plot of Barium

Iron

Figure 12: Probability plot of iron

Refractive index, Silicon, Aluminium, and Sodium are the only variables which are approximately normally distributed(Abdal-sahib et al., 2013; Tsokos et al., 2016).

1. Confidence intervals of Float and non-float glass

Table 1: Float Glass Confidence intervals for normally distributed variables

 Statistic Std. Error Refractive index Mean 1.5185708 .00023735 95% Confidence Interval for Mean Lower Bound 1.5180990 Upper Bound 1.5190426 Sodium Mean 13.2803 .05402 95% Confidence Interval for Mean Lower Bound 13.1730 Upper Bound 13.3877 Aluminum Mean 1.1711 .03080 95% Confidence Interval for Mean Lower Bound 1.1099 Upper Bound 1.2324 Silicon Mean 72.5772 .06030 95% Confidence Interval for Mean Lower Bound 72.4574 Upper Bound 72.6971

Table 2: Non-Float Glass Confidence intervals for normally distributed variables

 Statistic Std. Error Refractive index Mean 1.5186186 .00043613 95% Confidence Interval for Mean Lower Bound 1.5177497 Upper Bound 1.5194874 Sodium Mean 13.1117 .07618 95% Confidence Interval for Mean Lower Bound 12.9599 Upper Bound 13.2635 Aluminum Mean 1.4082 .03652 95% Confidence Interval for Mean Lower Bound 1.3354 Upper Bound 1.4809 Silicon Mean 72.5980 .08311 95% Confidence Interval for Mean Lower Bound 72.4325 Upper Bound 72.7636

References

Abdal-sahib, R., Altammar, S. M., Azlan, H. A., Aytemur, A., Balters, S., Steinert, M., … Canny, J. (2013). Testing for Normality. Frontiers in Psychology, 98(1), 1–8. https://doi.org/10.3389/fpsyg.2014.01470

Aspara, J. (2009). Aesthetics of stock investments. Consumption Markets & Culture, 12(2), 99–131. https://doi.org/10.1080/10253860902840917

Cohen, L., Manion, L., & Morrison, K. (2011). Descriptive Statistics. In Research methods in education (pp. 622–640). https://doi.org/10.1213/ANE.0000000000002471

Fletcher, J. (2009). Normal distribution. BMJ, 338(feb18 2), b646–b646. https://doi.org/10.1136/bmj.b646

Lynott, W. J. (2005). Stock investments. Water Well Journal, 59(12), 39. Retrieved from https://www.scopus.com/inward/record.url?eid=2-s2.0-30344484585&partnerID=40&md5=ba73235c348c82cf518a1f10cd84cbe4

McCluskey, A., & Lalkhen, A. G. (2007). Statistics III: Probability and statistical tests.s, 7(5), 167–170. https://doi.org/10.1093/bjaceaccp/mkm028

Titman, S., Wei, K. C. J., & Xie, F. (2004). Capital Investments and Stock Returns. Journal of Financial and Quantitative Analysis, 39(04), 677. https://doi.org/10.1017/S0022109000003173

Tsokos, C., Wooten, R., Tsokos, C., & Wooten, R. (2016). Normal Probability. In The Joy of Finite Mathematics (pp. 231–263). https://doi.org/10.1016/B978-0-12-802967-1.00007-3

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