Normal Distributions versus T-Distributions

Learning Outcomes

  • Determine which distribution (z or t) should be used for a given scenario
  • Verify assumptions for performing a hypothesis test have been met

Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. We perform tests of a population mean using a normal distribution or a Student’s t-distribution. (Remember, use a Student’s t-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.) We perform tests of a population proportion using a normal distribution (usually n is large or the sample size is large).

If you are testing a single population mean, the distribution for the test is for means:

[latex]\displaystyle\overline{{X}}\text{~}{N}{\left(\mu_{{X}}\text{ , }\frac{{\sigma_{{X}}}}{\sqrt{{n}}}\right)}{\quad\text{or}\quad}{t}_{{{d}{f}}}[/latex]

The population parameter is [latex]\mu[/latex]. The estimated value (point estimate) for [latex]\mu[/latex] is [latex]\displaystyle\overline{{x}}[/latex], the sample mean.

If you are testing a single population proportion, the distribution for the test is for proportions or percentages:

[latex]\displaystyle{P}^{\prime}\text{~}{N}{\left({p}\text{ , }\sqrt{{\frac{{{p}{q}}}{{n}}}}\right)}[/latex]

The population parameter is [latex]p[/latex]. The estimated value (point estimate) for [latex]p[/latex] is [latex]p'[/latex]. [latex]p' = \frac{x}{n}[/latex] where x is the number of successes and n is the sample size.

Recall: Order of Operations

When simplifying mathematical expressions, perform the operations in the following order:
1. Parentheses and other Grouping Symbols

  • Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

2. Exponents

  • Simplify all expressions with exponents.

3. Multiplication and Division

  • Perform all multiplication and division in order from left to right. These operations have equal priority.

4. Addition and Subtraction

  • Perform all addition and subtraction in order from left to right. These operations have equal priority.

[latex]\displaystyle{P}^{\prime}\text{~}{N}{\left({p}\text{ , }\sqrt{{\frac{{{p}{q}}}{{n}}}}\right)}[/latex]

To follow the order of operations and find the correct value, first find [latex]q = (1 - p)[/latex]. Then take that value and multiply by [latex]p[/latex]. Then divide by [latex]n[/latex]. Then lastly, take the square root.

Assumptions

When you perform a hypothesis test of a single population mean μ using a Student’s t-distribution (often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a t-test will work even if the population is not approximately normally distributed).

When you perform a hypothesis test of a single population mean μ using a normal distribution (often called a z-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.

When you perform a hypothesis test of a single population proportion p, you take a simple random sample from the population. You must meet the conditions for a binomial distribution which are as follows: there are a certain number n of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success p. The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np and nq must both be greater than five (np > 5 and nq > 5). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with μ = p and [latex]\displaystyle\sigma=\sqrt{{\frac{{{p}{q}}}{{n}}}}[/latex]. Remember that [latex]q = 1 – p[/latex].