Data analysis in business
Tutorial 1
Question A
Suppose that X1 ∼ N (1, 1), X2 ∼ N (1, 2), and X3 ∼ N (2, 1).
1. Sketch the pdf of X1 , X2 and X3 on the same graph.
2. Summarize how an increase in variance shifts the pdf.
3. Summarize how an increase in mean shifts the pdf.
4. Indicate on your figure the area representing Prob(X1 < 0) and Prob(X3 < 0).
5. Compare Prob(X1 < 0) and Prob(X3 < 0) on your graph. Which one is bigger?
Question B
Suppose that the age of real estate investors is normally distributed with mean equal to 40 years and
standard deviation 10 years.
1. What proportion of investors is below the age of 25 (express your answer in terms of the cdf of
a standard normal distribution)?
2. What proportion of investors is between the age of 40 and 60 (express your answer in terms of
the cdf of a standard normal distribution)?
3. What proportion of investors is older than 50 (express your answer in terms of the cdf of a
standard normal distribution)?
4. Find the age such that 90% of the investors are below this age (express your answer in terms of
the percentiles of a standard normal distribution).
5. Use Excel to obtain numerical answers to question B1-B4.
6. Suppose the mean age of investors is unknown but the standard deviation is 10 years. What is
the mean age of the investors if 5% of them are below 20 years (the 5th percentile of a standard
normal distribution is -1.645)?
Question C
The Excel file T1_WeeklyEarnings.xlsx contains weekly earnings of 856 individuals during August
2020 in Australia.
1. Use Excel to obtain the mean, median, and skewness. Comment on the skewness.
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2. Sort the data and find the first quartile.
3. Create a box-plot.
4. Create a histogram using the intervals partitioned by [0,250,500,750,1000,1250,1500,1750,2000,2250,
above 2250].
5. What percentage of earners made more than 2000 (not including 2000) per week?
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