Understand and use the concept that there are variations in confidence intervals between samples and that most but not all confidence intervals contain 𝜇.

A manufacturer measures the mass of ball bearings from production runs. The mass is known to be normally distributed with population standard deviation σ = 2.4 g. A quality control technician takes repeated random samples of 16 ball bearings each day and constructs a 95% confidence interval for the population mean mass μ each day over a 40-day period. (a) Calculate the margin of error for each daily confidence interval. (1 mark) (b) If the true population mean mass is μ = 24.8 g, approximately how many of the 40 confidence intervals would be expected to contain μ? (1 mark) (c) Explain why not all confidence intervals will contain the true population mean, even though the process is repeated under identical conditions. (1 mark)

A researcher takes 200 independent random samples of size 25 from a normally distributed population and calculates a 95% confidence interval for the population mean μ from each sample. The diagram shows the 200 confidence intervals plotted vertically, with the true population mean μ indicated by the vertical line. Which statement best describes the expected number of intervals that contain μ?

A manufacturing company measures the diameter (mm) of ball bearings produced by a machine. The diameter is normally distributed with an unknown population mean μ. The company takes repeated random samples of 25 ball bearings and calculates a 95% confidence interval for μ at the end of each production run. Over 200 production runs, approximately how many confidence intervals would be expected to fail to contain μ?

A school counsellor measures the weekly stress levels of Year 12 students using a standardised scale. The population mean is μ = 52 and the population standard deviation is σ = 8.5. The counsellor collects four random samples of size n = 36 and calculates a 95% confidence interval for μ from each sample. (a) Calculate the margin of error for a 95% confidence interval. (1 mark) (b) For each sample, determine the 95% confidence interval using the sample means provided in the stimulus. (2 marks) (c) State how many of the four confidence intervals contain the population mean μ = 52, and justify why not all confidence intervals necessarily contain μ. (1 mark)