STA10003 Foundations of Statistics Assignment 2 Part 1 (1)

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1 Name: Bianca Burrows Student ID: 104461782 Subject: STA10003: Foundations of Statistics Assignment 2: Analysis of data – Part 1 Due Date: 7 th January 2024

2 Question 1: The variable Birth indicates the birth order of each client Using SPSS, summarise the Birth variable and write a paragraph explaining the key features of the data observed in the output in the style presented in Reporting information about single variables (see supporting resources).

3 Participants Birth Order Frequency Percent Valid Percent Cumulative Percent Valid 1st child 452 45.2 50.2 50.2 2nd child 313 31.3 34.8 85.0 3rd child 110 11.0 12.2 97.2 4th child or higher 25 2.5 2.8 100.0 Total 900 90.0 100.0 Missing System 100 10.0 Total 1000 100.0 The ‘Birth Order’ of participants data reveals significant trends amongst the 1000 participants. The largest group in the sample is individuals identifying as the first-born child, comprising 42.5% of the study population. This suggests a high incidence of primogeniture in the research population. The smallest group is individuals who are the fourth or higher child, which accounts for 2.5% of the study population. This percentage reflects a smaller representation of individuals in higher birth order categories. The distribution of percentages among birth order categories emphasises the concentration of participants in the top three positions which account for 97.2% of the sample. Within the investigates population, such concentration shows a trend toward earlier birth orders.

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4 Question 2 The variable Height measures the height in cm for each client. Using SPSS, summarise the Height variable and write a paragraph explaining the key features of the data observed in the output in the style presented in Reporting information about single variables (see supporting resources).

5 Descriptives Statistic Std. Error Height (cm) Mean 168.6022 .33245 95% Confidence Interval for Mean Lower Bound 167.9498 Upper Bound 169.2547 5% Trimmed Mean 168.5515 Median 168.7000 Variance 99.910 Std. Deviation 9.99552 Minimum 139.40 Maximum 197.50 Range 58.10 Interquartile Range 15.12 Skewness .035 .081 Kurtosis -.469 .162

6 The height variable in the study has a mean height of 168.60cm, with a standard deviation of 9.99552cm, a variance of 99.910cm, and a range of 58.10cm. The interquartile range is 15.12cm and the 5% trimmed mean is 168.55cm. Skewness is minimal at 0.035, indicating a relatively symmetrical distribution. No outliers are observed, and the boxplot does not show data points beyond the whiskers. Overall, the height variable has a central tendency around the mean and median, a moderate spread of values, and minimal skewness, kurtosis, and no outlier. Question 3

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7 3 The variable AggInt measures the internal aggression [hostility] of each client. This variable was scored on a scale from 0 to 20, with higher scores representing greater levels of internal aggression [hostility]. Using SPSS, produce the relevant output to summarise the AggInt variable and write a paragraph explaining the key features of the data observed in the output in Reporting information about single variables (see supporting resources). Case Processing Summary Cases Valid Missing Total N Percent N Percent N Percent Internal Agression [Hostility] score 897 89.7% 103 10.3% 1000 100.0% Descriptives Statistic Std. Error Internal Agression [Hostility] score Mean 5.8852 .12546 95% Confidence Interval for Mean Lower Bound 5.6389 Upper Bound 6.1314 5% Trimmed Mean 5.6399 Median 5.0000 Variance 14.120 Std. Deviation 3.75761 Minimum 1.00

8 Maximum 20.00 Range 19.00 Interquartile Range 5.00 Skewness .975 .082 Kurtosis .503 .163 The ‘Internal Aggression (Hostility) score’ ranges from 1 to 20, with a median of 5.0000. In a boxplot, this median is at the centre of a box, representing the middle 50% of the scores (IQR = 5.00). The ‘whiskers’ extend to the minimum (1.00) and maximum (20.00) scores, showing the full range. The distribution slightly skews to the right (skewness =0.0975), visible as a loner tail on the higher scores side. It also has a moderately peaked shape (kurtosis = 0.503). This means most scores are concentrated around a median, with a few

9 higher values. In summary, the scores tend to be round 5.0000, with some higher values pulling the distribution slightly to the right, and a moderate peak indicating some concentration of scores. Question 4 In which competition was Susan’s performance better, relative to others who took part in that competition? Justify your answer, quoting relevant statistics as part of your explanation. Z=(x-m)/  M=119 X=125  =standard deviation Competition A: M=119 X=125  =7 Z=(x-m)/  (125-119)/7 =0.857

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10 Competition B: M=165 X=140  =15 Z=(x-m)/  =140-165/15 =-1.667 Competition C: m=109 X=118  =8 Z= x-m/  =118-109/8 =1.125 The z score is positive in competition A and C, which indicates Susan took the above the mean, however the Z score is negative in competition B, indicating she took time less than the mean time Therefore, in conclusion, Susan had better performance, relative to other participants of competition B. Question 5 a) What is the population we can draw conclusions about in this study? All the alkaline batteries manufactured by the Betterbuy Battery Company. b) What does the highlighted section of the distribution in Figure 1 represent?

11 It represents that there is a normal distribution; the sample mean will be the same for all possible sample sizes. c) The random sample of 625 batteries taken by the consumer organisation turned out to have a mean operating time of 20.16 hours. Does this sample look like it belongs to the sampling distribution displayed in Figure 1? Justify your answer. Sample mean: 20.16 hours N=625 Z = 20.16−201.5√625 = 2.67 Z=20.16−201.5√625= 0.16 / (1.5 / 25) = 0.16 / 0.06 = 2.67 Z = 2.67 The Z-score of 2.67 fits within this range, and the sample mean of 20.16 hours is not within a significant distance of the expected mean (20.00 hours) in the sampling distribution. Therefore, this calculation further supports the conclusion that the sample mean of 20.16 hours belongs to the sampling distribution, as it falls within the expected range of values for a normal distribution which is -3 to +3. d) Given that the sample was randomly selected from all batteries manufactured by the Betterbuy Battery Company in March, and given that each battery’s operating time was measured accurately, what conclusion can we reach from part (c)?

12 From part C, we can conclude that Betterbuy Battery Company’s claim is correct and that the mean operating time of batteries is less than 20 hours.

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STA10003 Foundations of Statistics Assignment 2 Part 1 (1)

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