Understand and use the relationship between margin of error, level of confidence and sample size.

A survey is conducted to estimate the mean weekly expenditure on groceries for households in a region. A sample of 64 households has a mean expenditure of $\(\$156.40\) with a standard deviation of $\(\$28.00\). (a) Calculate the margin of error at a 95% confidence level. Give your answer to the nearest cent. (1) (b) If the researchers want to reduce the margin of error to $\(\$3.50\) while maintaining the same confidence level and standard deviation, determine the minimum sample size required. (1) (c) Explain why increasing the sample size reduces the margin of error. (1)

A research team is investigating the average daily screen time of teenagers in a region. They plan to construct a 95% confidence interval for the population mean based on a random sample. The population standard deviation is known to be 1.8 hours. (a) Calculate the margin of error if the sample size is 64 teenagers. (1 mark) (b) Determine the sample size required to reduce the margin of error to 0.3 hours at the same 95% confidence level. (1 mark) (c) The team decides to increase the confidence level to 99% while keeping the sample size from part (b). Calculate the new margin of error, correct to 2 decimal places. (1 mark)

A market research team estimates the average spending of customers using a confidence interval based on a sample of 64 customers, with a margin of error (half-width) of \(±$2.40\). The same population standard deviation is used, and the team wishes to produce a confidence interval with a margin of error of \(±$0.60\) using a new sample. (a) Explain what must happen to the sample size to achieve the smaller margin of error. (1 mark) (b) Calculate the new sample size required. (2 marks)