Use repeated random sampling data from a variety of distributions and a range of sample sizes to examine properties of the distribution of 𝑋 across samples of a fixed size 𝑛, including its mean 𝜇, its standard deviation 𝜎 √𝑛 (where 𝜇 and 𝜎 are the mean and standard deviation of 𝑋) and its approximate normality if 𝑛 is large.

A logistics company records the delivery times (in hours) for packages across a large region. The delivery times are uniformly distributed with a minimum of 2 hours and a maximum of 14 hours. The company takes random samples of 36 packages. (a) Calculate the mean and standard deviation of the delivery times for individual packages. (1 mark) (b) Calculate the mean and standard deviation of the distribution of sample means for samples of size 36. (2 marks) (c) Explain why the distribution of sample means would be approximately normal despite the population distribution being uniform. (1 mark)

A manufacturing plant produces resistors with resistance values that follow a uniform distribution with a mean of 1,000 Θ and a standard deviation of 115 Θ. Quality control staff randomly sample 36 resistors and calculate the mean resistance for each sample. Over many repeated samples of this size, use this information to: (a) Determine the expected mean of the distribution of sample mean resistances. (1 mark) (b) Calculate the standard deviation of the distribution of sample mean resistances. (1 mark) (c) Explain, with reference to the central limit theorem, why the distribution of sample means can be assumed to be approximately normally distributed despite the population distribution being uniform. (2 marks)

A pharmaceutical company is testing the effectiveness of a new medication. The time taken (in minutes) for the medication to reach peak effectiveness in a population has a mean of 45.2 minutes and a standard deviation of 8.6 minutes. The distribution of times in the population is known to be skewed with positive skewness. The company conducts repeated random sampling from this population. (a) Explain why the distribution of sample means from samples of size n = 120 will be approximately normally distributed, despite the population distribution being skewed. (1 mark) (b) Calculate the standard deviation of the distribution of sample means for samples of size n = 120. Give your answer correct to 2 decimal places. (1 mark) (c) A random sample of size n = 120 yields a sample mean of 44.8 minutes. Calculate the probability that a randomly selected sample mean from samples of size n = 120 will be less than or equal to 44.8 minutes. Give your answer correct to 4 decimal places. (2 marks) (d) The company wishes to reduce the standard deviation of the sample mean distribution to 0.60 minutes by increasing the sample size. Determine the required sample size. Give your answer to the nearest whole number. (1 mark)

A manufacturing process produces ball bearings whose diameter (in millimetres) follows a distribution with mean μ = 12.50 and standard deviation σ = 0.84. An engineer carries out repeated random sampling, recording the mean diameter for samples of various sizes. The results for samples of size 16 and size 64 are shown in the table below. (a) For each sample size, calculate the theoretical standard deviation of the sample mean distribution. (2 marks) (b) The engineer observes that 48 of the 120 samples of size 16 have a sample mean greater than 12.71 mm. Estimate the probability that a sample of size 16 has a sample mean greater than 12.71 mm. (1 mark) (c) Explain why the distribution of sample means can be assumed to be approximately normal for both sample sizes, and use this assumption to calculate the probability that a sample of size 64 has a sample mean greater than 12.71 mm. Give your answer correct to 3 significant figures. (2 marks)

A company manufactures ball bearings whose diameters follow a right-skewed distribution with mean μ = 24.8 mm and standard deviation σ = 2.1 mm. Quality control staff take repeated random samples of 36 ball bearings and record the mean diameter for each sample. (a) State the mean and standard deviation of the distribution of sample means. (1 mark) (b) Explain why the distribution of sample means can be assumed to be approximately normal, even though the population distribution is not normal. (1 mark) (c) Use the properties of the distribution of sample means to determine the probability that a random sample of 36 ball bearings will have a mean diameter greater than 25.3 mm. (1 mark)

A machine fills bottles with liquid. The volume of liquid in each bottle (in mL) follows a distribution with mean $\mu = 500$ mL and standard deviation $\sigma = 8$ mL. A quality control inspector takes repeated random samples of 25 bottles and measures the mean volume $\bar{X}$ for each sample. (a) Calculate the mean of the sampling distribution of $\bar{X}$. (1 mark) (b) Calculate the standard deviation of the sampling distribution of $\bar{X}$. (1 mark) (c) Explain why the sampling distribution of $\bar{X}$ can be assumed to be approximately normal, even if the underlying population distribution is not normal. (1 mark)

A manufacturing plant produces metal washers whose diameter is approximately normally distributed with population mean μ = 24.8 mm and population standard deviation σ = 0.62 mm. A quality-control team collects repeated random samples of washers and calculates the mean diameter for each sample. (a) If samples of size n = 16 are used, calculate the standard deviation of the distribution of the sample mean diameter. (1) (b) The quality-control team notices that when using samples of size n = 16, approximately 5% of the sample means fall below 24.65 mm. Use the standard deviation from part (a) to justify whether this observation is reasonable. (2) (c) State what happens to the standard deviation of the distribution of sample means when the sample size is increased from n = 16 to n = 64, and explain how this demonstrates the effect predicted by the Central Limit Theorem. (1)

A pharmaceutical company tests the effectiveness of a new tablet. The active ingredient concentration X (in mg) in tablets from a large batch is known to follow a distribution with mean μ = 498 mg and standard deviation σ = 22 mg. The company conducts quality control by repeatedly selecting random samples of size n and calculating the mean concentration of each sample. (a) Calculate the standard deviation of the distribution of sample means, σₓ̄, if samples of size n = 64 are taken. (1 mark) (b) Given that sample means follow an approximately normal distribution, calculate the probability that a single sample mean from n = 64 tablets lies between 494 mg and 502 mg. (2 marks) (c) If the company wishes to reduce the standard deviation of the sample mean distribution to 2.2 mg, what sample size must be used? (2 marks)