Exercises: Margin of Error and Hypothesis Tests

A random sample of $100$ people is taken.

1. a) Use the formula $\dfrac{1}{\sqrt{n}}$ to find the margin of error as a percentage.

1. b) Find the margin of error for a sample of size $400$.

In a random sample of $200$ students, $120$ said they travel to school by bus.

Find the sample proportion, as a percentage.

A poll of $625$ voters finds that $48\%$ support a proposal.

Find the margin of error and write down the interval within which the population proportion is likely to lie, to one decimal place.

A survey of $900$ randomly selected adults in Ireland found that $54\%$ own a bicycle.

1. a) Find the margin of error as a percentage, correct to one decimal place.

1. b) Find the $95\%$ confidence interval for the population proportion.

1. c) A newspaper claims that the majority of adults own a bicycle. Is this supported by the interval? Explain.

A local council wants a margin of error of no more than $2\%$ in a poll.

1. a) Find the smallest sample size required.

1. b) The council instead samples $1{,}600$ people. Find the margin of error.

A coin is tossed $100$ times and lands on heads $59$ times. A student wishes to test whether the coin is fair.

1. a) State the null hypothesis.

1. b) Find the margin of error and the interval around the sample proportion.

1. c) Decide whether to accept or reject the null hypothesis at the $5\%$ level, and give a reason.

A manufacturer in Waterford claims that $90\%$ of its light bulbs last more than $1{,}000$ hours. In a random sample of $400$ bulbs, $351$ last more than $1{,}000$ hours.

1. a) State the null hypothesis and the alternative hypothesis.

1. b) Find the sample proportion and the margin of error.

1. c) Test the claim at the $5\%$ level of significance, stating your conclusion in context.

Two polls on a local election are carried out. Poll A has $400$ people and shows $46\%$ support for a candidate. Poll B has $1{,}600$ people and shows $44\%$ support.

1. a) Find the margin of error for each poll.

1. b) Explain whether the results of the two polls are consistent with each other.

1. c) Explain why Poll B is more reliable, referring to sampling variability.

A candidate says that $50\%$ of voters in her constituency support her. In a random sample of $n$ voters, $56\%$ support her.

1. a) Find the smallest sample size $n$ for which the claim of $50\%$ would be rejected at the $5\%$ level of significance, correct to the nearest whole number.

1. b) Explain why a smaller sample would not allow the claim to be rejected.

1. c) Explain what is meant by a Type I error in this context.