An SRS of families is selected from all Ontario families and family income is re

Question

An SRS of families is selected from all Ontario families and family income is recorded for each family for the year 2014. In Ontario, family incomes possess a moderately skewed distribution. If we select an SRS of 400 families, which of the following statements are valid conclusions that may be properly justified by the Central Limit Theorem? I) The histogram for the sample of 400 incomes will be approximately normal II) The mean income for the sample will be approximately normally distributed III) The standard deviation for the sample will be approximately normally distributed IV) The number of families in the sample will be approximately normally distributed V) The proportion of families for which the income is greater than exist100,000 will be approximately normally distributed A) All statements or no statements are correct B) Exactly four statements or exactly one statement is correct C) I and IV D) II and V E) I, II and V

Explanation / Answer

The central limit theorem states that given a large enough sample their sum or averages tends to have a normal distribution. This same for proportions too .

Therefore based on this

the correct statements are

II The mean income for the sample will be normally distributed.

V The proportion of families for which the income is greater than $100,000 will be approximately normally distributed.

Hence the correct option would be D) II and V

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