Use 𝑥̅ and 𝑠 to estimate 𝜇 and 𝜎, to obtain approximate intervals covering desired proportions of values of a normal random variable and compare with an approximate confidence interval for 𝜇.

The diameter (mm) of steel rods produced by a factory is normally distributed. A random sample of 50 rods has a mean diameter of 18.3 mm and a sample standard deviation of 0.64 mm. What is the approximate 95% confidence interval for the population mean diameter?

The battery life of a particular brand of smartphone is known to be normally distributed. A sample of 18 smartphones is randomly selected, and the battery life (in hours) is measured for each. The sample produces a mean of x̄ = 48.3 hours and a sample standard deviation of s = 3.7 hours. (a) Using the normal approximation with z = 1.96, calculate an approximate 95% confidence interval for the mean battery life μ. (2 marks) (b) The manufacturer claims that 90% of batteries last between a and b hours, where these values are symmetric about the mean. Use the sample statistics to estimate the population mean and population standard deviation, and hence calculate the values of a and b. (2 marks)

A food processing company monitors the net mass of canned soup. The manufacturer claims the net mass is normally distributed with a mean of 415 g. A random sample of 40 cans is measured, yielding a sample mean of x̄ = 412.8 g and sample standard deviation of s = 8.2 g. (a) Use the sample statistics to construct a 95% approximate confidence interval for the population mean μ. (2 marks) (b) The manufacturer also wishes to estimate the range within which approximately 90% of individual can masses lie. Using the sample statistics to estimate the population parameters, determine the limits of this approximate prediction interval. (2 marks) (c) Does the claimed population mean of 415 g lie within the 95% confidence interval calculated in part (a)? Explain what this suggests about the manufacturer's claim. (1 mark)

A coffee shop measures the daily sales revenue from their espresso machine. A random sample of 50 trading days yields a sample mean of $385.60 and a sample standard deviation of $42.30. Assume the daily revenue is normally distributed. (a) Use the sample statistics to estimate the population mean μ and population standard deviation σ. (b) Construct an approximate 95% confidence interval for μ. (c) Calculate the interval that would capture approximately 95% of all individual daily sales values, and explain how it differs from the confidence interval in part (b).