Understand and use the approximate confidence interval (𝑥̅ − 𝑧 𝑠 √𝑛, 𝑥̅ + 𝑧 𝑠 √𝑛), as an interval estimate for 𝜇, the population mean, where 𝑧 is the appropriate quantile for the standard normal distribution.

A telecommunications company measures the data usage (in GB per month) of a random sample of 50 customers. The sample mean is 18.4 GB and the sample standard deviation is 3.7 GB. (a) Calculate an approximate 90% confidence interval for the population mean data usage. Give your answer to two decimal places. (b) Calculate an approximate 95% confidence interval for the population mean data usage. Give your answer to two decimal places. (c) Explain why the 95% confidence interval is wider than the 90% confidence interval.

A retail chain measures the daily takings (in dollars) from a random sample of 50 stores. The sample mean is $3,847 with a standard deviation of $312. (a) Calculate an approximate 90% confidence interval for the population mean daily takings. Give your answer to the nearest dollar. (2 marks) (b) Calculate an approximate 95% confidence interval for the population mean daily takings, using the same sample data. Give your answer to the nearest dollar. (2 marks)

A local council surveys the annual household water consumption in a suburb. A random sample of 64 households was selected, and the sample mean consumption was found to be 487.3 kL with a sample standard deviation of 38.6 kL. (a) Determine an approximate 95% confidence interval for the population mean annual household water consumption. Give your answer to one decimal place. (2 marks) (b) The council claims that the mean annual household water consumption is 475 kL. Evaluate the reasonableness of this claim using your confidence interval and mathematical reasoning. (2 marks)

A marine biologist measures the shell diameter of a random sample of 64 sea urchins from a coastal population. The sample mean is 32.8 mm and the sample standard deviation is 4.2 mm. (a) Determine an approximate 90% confidence interval for the population mean shell diameter. (2 marks) (b) Explain why the 95% confidence interval would be wider than the 90% confidence interval, using mathematical reasoning. (2 marks)

A manufacturing company records the tensile strength (in MPa) of steel samples from a production run. A random sample of 18 samples was tested, yielding a sample mean of 487.3 MPa and a sample standard deviation of 23.8 MPa. The tensile strength is assumed to be normally distributed. Calculate a 90% confidence interval for the population mean tensile strength, μ. Give your answer to one decimal place.

A recruitment agency records the time taken (in minutes) for job applicants to complete an online assessment. A random sample of 50 applicants had a mean completion time of 23.7 minutes and a sample standard deviation of 3.2 minutes. (a) Calculate an approximate 90% confidence interval for the mean completion time of all applicants. Give your answer to one decimal place. (b) Explain what the 90% confidence level means in this context.

A school's network administrator records the loading time for a standard application across a random sample of 50 computers. The sample mean loading time is 2.34 seconds with a sample standard deviation of 0.48 seconds. (a) Calculate an approximate 90% confidence interval for the population mean loading time. Give your answer to two decimal places. (b) The administrator believes the true population mean loading time is 2.15 seconds. Use your confidence interval from part (a) to comment on the reasonableness of this belief. Justify your answer.

The monthly electricity consumption (in kWh) for households in a regional area is assumed to be normally distributed. A random sample of 35 households yielded a sample mean of 487.3 kWh and a sample standard deviation of 62.8 kWh. (a) Determine an approximate 90% confidence interval for the population mean monthly electricity consumption. Give your answer to one decimal place. (2 marks) (b) Another researcher, using a different random sample from the same population, calculates a 95% confidence interval of (451.2, 509.8) kWh. Determine the sample size used in this second study, given that the population standard deviation is 78.4 kWh. Give your answer to the nearest whole number. (3 marks)

A marine biologist collects data on the shell diameter of a population of sea urchins. A random sample of 36 urchins is measured, yielding a sample mean of 42.7 mm and a sample standard deviation of 3.8 mm. (a) Calculate an approximate 90% confidence interval for the population mean shell diameter. Give your answer to one decimal place. (2 marks) (b) Calculate an approximate 99% confidence interval for the population mean shell diameter. Give your answer to one decimal place. (2 marks)

A pharmaceutical company is testing the effectiveness of a new drug by measuring the time (in minutes) it takes for the drug to reach peak concentration in the bloodstream. A random sample of 50 patients was selected, yielding a sample mean of 28.7 minutes and a sample standard deviation of 4.53 minutes. Assume the time to peak concentration is normally distributed. (a) Determine a 90% confidence interval for the population mean time to peak concentration. Give your answer to two decimal places. (2 marks) (b) A different random sample of 35 patients was taken from the same population. The 95% confidence interval calculated from this sample was (26.84, 29.56) minutes. Calculate the sample mean and sample standard deviation for this second sample, giving your answers to one decimal place. (3 marks)

A dietary research team measures the daily sodium intake (in milligrams) of a random sample of adults from a particular region. The data is summarised in the table in the stimulus. (a) Calculate the sample mean and sample standard deviation using the class midpoints. Give your answers to 1 decimal place. (2 marks) (b) Calculate an approximate 98% confidence interval for the population mean daily sodium intake. Give your answer to the nearest milligram. (2 marks) (c) A health organisation claims that the mean daily sodium intake in this region is 2,100 mg. Evaluate the reasonableness of this claim using your confidence interval from part (b). Justify your reasoning. (1 mark)