Use repeated random sampling data from a variety of distributions and a range of sample sizes to examine the approximate standard normality of 𝑋−𝜇 𝑠 √𝑛 for large samples (𝑛 ≥ 30), where 𝑠 is the sample standard deviation (Central limit theorem).

A researcher is investigating the Central Limit Theorem (CLT) by generating 10 000 random samples of size n = 50 from an exponential distribution with population mean μ = 8 minutes. The stimulus shows a histogram of the standardised sample means calculated as (X̄ − μ)/(s/√n), where s is the sample standard deviation for each sample. a) One particular sample has sample mean X̄ = 9.4 minutes and sample standard deviation s = 8.2 minutes. Calculate the standardised sample mean for this sample, correct to 2 decimal places. (1 mark) b) State the mean and standard deviation of the theoretical distribution that the histogram should approximate, according to the Central Limit Theorem. (1 mark) c) Explain why the Central Limit Theorem applies in this situation even though the original population distribution is exponential (not normal). (1 mark)

A population has mean μ = 85 and standard deviation σ = 12. A researcher repeatedly samples from this population, taking samples of size n = 50 and calculating the mean of each sample. Calculate the approximate standard deviation of the sample mean distribution.

A large population has an unknown distribution with population mean μ = 48.5 and population standard deviation σ = 7.2. Repeated random samples of size n = 50 are drawn from this population. The sample means are recorded and their standard deviation is found to be approximately 1.02. (a) Calculate the theoretical standard error of the sample mean and compare it to the observed standard deviation of the sample means. (2 marks) (b) A single new sample of size n = 50 yields a sample mean of χ = 50.1. Use the Central Limit Theorem to determine the approximate probability that a sample mean is at least this value. (2 marks)

A random variable (X) is known to follow an exponential distribution with mean (μ = 8.5). Random samples of various sizes are repeatedly drawn from this distribution. The table below records the sample means and sample standard deviations from 40 samples of each size. Using the standardised test statistic (Z = (X̄−μ)/(s/√n)), determine: (a) The standardised test statistic for the sample of size (n = 30) with sample mean (X̄ = 8.92) and sample standard deviation (s = 8.17). ([2 marks]) (b) The standardised test statistic for the sample of size (n = 100) with sample mean (X̄ = 8.31) and sample standard deviation (s = 8.04). ([1 mark]) (c) By comparing these statistics, explain what the results suggest about the behaviour of the standardised distribution as sample size increases, with reference to the Central Limit Theorem. ([2 marks])