4.1 Sampling Distributions for Sample Means
- Syllabus
- 2026
- Topic
- 4.1
- Level
- —
Learning objectives
4.1A—Calculate the mean and standard deviation of a sampling distribution of a sample meanCalculate the mean and standard deviation of a sampling distribution of a sample mean.• For a population with population mean µ and population standard deviation σ, when the sampled values are independent, the sampling distribution of the sample mean has mean μμx = and standard deviation σσx = n .4.1B—Justify the appropriateness of conditions for the sampling distribution of a sample meanJustify the appropriateness of conditions for the sampling distribution of a sample mean.• Sampling without replacement requires that two conditions must be met:- i. The randomization condition—the data should be collected using a random sample.- ii. The 10% condition—the population size must be at least 10 times larger than the sample size (%nN≤10 ), where N is the size of the population and n is the sample size.• For a quantitative variable, if the population distribution can be modeled by a normal distribution, the sampling distribution of the sample mean, x, can be modeled with a normal distribution regardless of the sample size.• For a quantitative variable, if the population distribution cannot be modeled by a normal distribution, the sampling distribution of the sample mean, x, can be modeled approximately by a normal distribution, provided n ≥30. If the population distribution is extremely skewed, a sample size much larger than 30 may be needed to ensure the sampling distribution is approximately normal.4.1C—Interpret the mean, standard deviation, and probabilities for the sampling distribution of a sample meanInterpret the mean, standard deviation, and probabilities for the sampling distribution of a sample mean.• The mean, standard deviation, and probabilities for a sampling distribution of a sample mean should be interpreted within the context of a specific population. 122 Inference for Quantitative Data: Means UNIT 4
4.1.A—Calculate the mean and standard deviation of a sampling distribution of a sample mean
Calculate the mean and standard deviation of a sampling distribution of a sample mean.
- For a population with population mean µ and population standard deviation σ, when the sampled values are independent, the sampling distribution of the sample mean has mean μμx = and standard deviation σσx = n .
4.1.B—Justify the appropriateness of conditions for the sampling distribution of a sample mean
Justify the appropriateness of conditions for the sampling distribution of a sample mean.
- Sampling without replacement requires that two conditions must be met:
- i. The randomization condition—the data should be collected using a random sample.
- ii. The 10% condition—the population size must be at least 10 times larger than the sample size (%nN≤10 ), where N is the size of the population and n is the sample size.
- For a quantitative variable, if the population distribution can be modeled by a normal distribution, the sampling distribution of the sample mean, x, can be modeled with a normal distribution regardless of the sample size.
- For a quantitative variable, if the population distribution cannot be modeled by a normal distribution, the sampling distribution of the sample mean, x, can be modeled approximately by a normal distribution, provided n ≥30. If the population distribution is extremely skewed, a sample size much larger than 30 may be needed to ensure the sampling distribution is approximately normal.
4.1.C—Interpret the mean, standard deviation, and probabilities for the sampling distribution of a sample mean
Interpret the mean, standard deviation, and probabilities for the sampling distribution of a sample mean.
- The mean, standard deviation, and probabilities for a sampling distribution of a sample mean should be interpreted within the context of a specific population. 122 Inference for Quantitative Data: Means UNIT 4