Answer all of the following questions.
1. A sample of Irish households shows the following size distribution:
|
Household Size
|
Frequency
|
|
1
|
33
|
|
2
|
41
|
|
3
|
26
|
|
4
|
24
|
|
5
|
14
|
|
6
|
5
|
|
7
|
1
|
|
8
|
1
|
(a) Calculate the mean, median and standard deviation of household size.
(b) Draw a histogram of household size variable. On the histogram, mark in the mean and median. Comment on the shape of the distribution.
(c) How is an ‘outlier' defined? Is an observation of a household size of 8 an outlier?
2. A survey of Irish people found that 21.3% of the sample had green eyes. Say that 12 people (all Irish) are in a bar, and one of them tells the others about this fascinating research on eye colour. What is the probability that three of the 12 has green eyes? What is the probability that none has green eyes? Explain the calculations that you make carefully.
3. A survey of 100 households drawn randomly from the population found that households in the sample spent an average of €949 over the Christmas season. The sample standard deviation was €175.
(a) What estimator would you use to estimate the population mean Christmas spending? What is meant by an unbiased estimator? Is your estimator unbiased? Explain.
(b) If Christmas spending is normally distributed, what is the probability that a randomly selected household spends between €750 and €1150?
(c) Say that you are asked to construct the sampling distribution of mean Christmas spending. Give a step-by-step explanation of how you would do this; you can assume a sample size of 100.
(d) Sketch (roughly) the sampling distribution of mean Christmas spending. Explain why the sampling distribution has this shape. If Christmas spending had a skewed distribution (rather than normal), would this change your sketch of the sampling distribution? Explain.
(e) What are the mean and standard error of your sampling distribution?