Conditions for Inference about a Mean

Conditions for Inference about a Mean

• The text is about using statistical inference to draw conclusions about a population mean.
• The most common scenario is when the population standard deviation is unknown - This is the most realistic situation.
• To test for a mean, you need these conditions:
  • Random Sample: This is crucial. The sample must be representative of the population you are interested in.
  • Normal Distribution: The population must be normally distributed.
    • Large sample size: If your sample size is large enough, the Central Limit Theorem guarantees that the sample mean will be approximately normally distributed, regardless of the population distribution.
    • The minimum sample size should be 30.
  • Population Standard Deviation: The population standard deviation should be known. In reality, it is often not, so we need to estimate it.

Standard Error for Unknown σ

• When the population standard deviation (σ) is unknown, we need to estimate it using the sample standard deviation (s)
• This estimate, s, is used to calculate the standard error of the mean which is the standard deviation of the sampling distribution of the sample mean.
• The formula for the standard error of the mean (SEM) is: SEM = s / √n

One-sample t statistic

• The t statistic is used for inference about a mean, especially when the population standard deviation (σ) is unknown, and it is estimated by the sample standard deviation (s).
• It is used with a t-distribution instead of a normal distribution.
• It is calculated as: (sample mean - hypothesized mean) / SEM

Properties of t-distribution

• The t-distribution is a bell-shaped distribution, similar to the standard normal distribution.
• It is a family of distributions - many different t-distributions, and the specific one used depends on the degrees of freedom.
• Degrees of Freedom: The degrees of freedom (df) are determined by the sample size (n). For one-sample t-test, the degrees of freedom are n-1.
• Comparison with the Standard Normal Distribution:
  • The t-distribution has a slightly higher variance (it is more spread out) than the standard normal distribution.
  • As the sample size (n) gets larger, the t-distribution gets closer to the standard normal distribution.
    • This is because with larger sample size, our estimate s converges more closely to σ.

Conditions when using the t-procedures

• Small Samples (n < 15):
  • The t procedures can be used if the data appear close to normal (symmetric, single peak, no outliers). However, avoid using t if the data are clearly skewed, or if outliers are present.
• Medium Samples (n ≥ 15):
  • The t procedures can be used unless there is evidence of strong skewness or outliers.
• Large Samples (n ≥ 40):
  • The t procedures can be used even for clearly skewed distributions, as long as the sample size is large.
  • The larger the sample size, the more robust the t-procedures to departures from normality.

Examples

• Example 1: A histogram of the percentage of Hispanic residents in each US state.
  • This is a census data, not a random sample. Therefore, t-procedures cannot be used.
• Example 2: Using a stem plot to show the force required to pull apart 20 randomly selected pieces of Douglas fir.
  • The data may be skewed right, but the t-procedures can still be used.
• Example 3: A histogram showing the word lengths of 80 random words in a sample from Shakespeare’s plays.
  • It is not a random sample, and therefore the t-procedures are not appropriate.

Conditions for Inference about a Mean

• Requires a random sample from the population of interest.
• The population must have a Normal distribution or a large sample size (Central Limit Theorem) if the population is skewed.
• The population standard deviation (σ) is unknown.

Standard Error for Unknown σ

• Used when the population standard deviation (σ) is unknown.
• Estimated by the sample standard deviation (s).

One-sample t statistic

• A test statistic used to test hypotheses about the population mean when the population standard deviation is unknown.
• Calculated by dividing the difference between the sample mean and the hypothesized population mean by the standard error.

Properties of t-distribution

• Similar in shape to the standard Normal curve, symmetric about 0, and bell-shaped.
• More variability than the standard Normal curve.
• Approximates the standard Normal distribution as the sample size (n) gets larger.

Conditions for Using t-procedures

• Random sample from the population of interest is crucial, especially for small samples.
• The t-procedure can be used for different sample sizes:
  • Small Samples (n < 15): Use t-procedures if the data appear close to Normal (symmetric, single peak, no outliers). Avoid using t if the data are skewed or have outliers.
  • Medium Samples (n ≥ 15): Use t-procedures unless there are outliers or strong skewness in the data.
  • Large Samples (n ≥ 40): Use t-procedures even for clearly skewed distributions.

Example

• t-procedure can be used for the force required to pull apart 20 random pieces of Douglas fir because the data is skewed right but the sample size is sufficient.
• t-procedure can not be used for the distribution of word lengths in a sample of Shakespeare's plays as the sample size is too small and the distribution is unknown.
• t-procedure cannot be used for the histogram showing the percent of each state's residents who are Hispanic in the USA because this is census data rather than a random sample.