Understand the concept of the sample mean 𝑋 as a random variable whose value varies between samples where 𝑋 is a random variable with mean 𝜇 and the standard deviation 𝜎.

A production facility manufactures electrical components. The resistance of each component is normally distributed with a population mean of μ = 500 ohms and a population standard deviation of σ = 45 ohms. a) Calculate the standard error of the sample mean X when a random sample of i = 36 components is selected. (1 mark) b) If 50 different random samples of 36 components are collected, explain why the sample means X will differ from one another even though they come from the same population. (1 mark) c) Determine the probability that a sample mean lies between 490 and 510 ohms for samples of size i = 36. (1 mark)

A manufacturer measures the diameter of ball bearings produced by a machine. The diameter of all ball bearings in the population has a mean of μ = 24.5 mm and a standard deviation of σ = 0.8 mm. Refer to the table showing three different random samples. (a) Calculate the mean and standard deviation of the sampling distribution of the sample mean \(\bar{X}\) if samples of size \(n = 16\) are repeatedly drawn from this population. (1 mark) (b) If samples of size \(n = 64\) are drawn instead, calculate the new standard deviation of the sampling distribution and explain why this change occurs. (2 marks)

A manufacturer records the daily production output (in units) of a factory machine. The output X is normally distributed with a population mean of μ = 850 units and a population standard deviation of σ = 60 units. Each week, a random sample of 25 daily production records is selected and the sample mean X̄ is calculated. (a) Explain why the sample mean X̄ is a random variable. (1) (b) State the mean and standard deviation of the distribution of sample means. (1)

A bottling company fills bottles with mineral water. The volume of water in each bottle is normally distributed with mean μ = 502 mL and standard deviation σ = 8 mL. (a) State the mean and standard deviation of the distribution of the sample mean ̄X when samples of size n = 16 are taken. (1 mark) (b) Calculate the standard deviation of ̄X when the sample size is increased to n = 64. (1 mark) (c) Explain why the standard deviation of the sample mean decreases as the sample size increases. (1 mark)

A large population of students has a mean study time of μ = 8.5 hours per week with a standard deviation of σ = 2.3 hours. A researcher repeatedly selects random samples of size n = 36 from this population and records the sample mean study time Ā for each sample. (a) Determine the mean and standard deviation of the sampling distribution of Ā. (2) (b) Hence, determine the probability that a randomly selected sample mean lies between 8.0 and 9.0 hours per week. (2)