4.2 Constructing a Confidence Interval for a Population Mean

Describe t-distributions.

• t-distributions, also called Student’s t-distributions, form a family of symmetric, bell-shaped, standardized distributions with wider tails than that of the standard normal distribution. Specific t-distributions are identified using a parameter known as the number of degrees of freedom (df), which is based on the sample size(s). When the degrees of freedom are small, the t-distribution has a much narrower peak and fatter tails than a normal distribution. As the degrees of freedom increase, the t-distribution more closely resembles the standard normal distribution (mean μ = 0 and standard deviation σ =1).
• t-distributions are used for finding critical values and test statistics for inferences about a population mean, µ, when the population standard deviation, σ, is unknown and the sample standard deviation, s, must be used instead.

Identify an appropriate confidence interval procedure including the parameter for a population mean or population mean difference.

• The appropriate confidence interval procedure for estimating the population mean of a quantitative variable for one sample is a one-sample t-interval for a population mean. (The population standard deviation, σ, is not typically known for distributions for quantitative variables.)
• For a matched pairs design with two dependent samples, the appropriate analysis calculates differences between pairs of values to produce one sample of differences. The confidence interval procedure for the matched pairs design is a one-sample t-interval for a population mean difference.
• The parameter for a confidence interval for a population mean or population mean difference should reference the population mean or population mean difference and the response variable, in context. For the population mean difference, it is important to state the order of subtraction for the difference.

Justify the appropriateness of constructing a confidence interval for a population mean or population mean difference by verifying conditions.

• A one-sample t-interval for a population mean or population mean difference requires that three conditions be met:
  • i. The randomization condition—the data should be collected using a random sample or a randomized experiment.
  • ii. The 10% condition—when sampling without replacement, 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.
  • iii. The sample data condition—it is indicated the population distribution is approximately normal, or n ≥30, or if n <30, the sample data distribution should be free from strong skewness and outliers. For matched pairs, the number of differences should be greater than or equal to 30. If the number of differences is less than 30, the sample of differences should be free from strong skewness and outliers. 124 Inference for Quantitative Data: Means UNIT 4

Calculate an appropriate confidence interval for a population mean or population mean difference.

• A point estimate for a population mean is the sample mean, x, or xd for the sample mean difference.
• To estimate the population mean for one sample or the population mean difference between values in matched pairs, when the population standard deviation is unknown, the confidence interval is * sx±t n , where t∗ is the critical value for the central C% of a t-distribution with degrees of freedom n −1.

Calculate the standard error and margin of error for a sample size for a one-sample t-interval.

• The standard error (SE) for a sample mean is given by sSEx = n .
• For a one-sample t-interval for a population mean, the margin of error is the critical value (t∗) times the standard error (SE ), which equals ( * st ){ } | |{ n } . 125 Inference for Quantitative Data: Means UNIT 4