• Check the conditions for a [latex]t[/latex]-distribution, then use a [latex]t[/latex]-distribution to calculate probabilities when appropriate.

Normal Distribution vs. [latex]t[/latex]-distribution

Particular distributions are associated with hypothesis testing. We perform tests of a population mean using a normal distribution or a [latex]t[/latex]-distribution.

Remember, use a [latex]t[/latex]-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.

When you perform a hypothesis test of a single population mean [latex]\mu[/latex] using a [latex]t[/latex]-distribution (often called a [latex]t[/latex]-test), there are fundamental assumptions that need to be met 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 [latex]t[/latex]-test will work even if the population is not approximately normally distributed).

Remember, use a (normal) [latex]z[/latex]-distribution when the population standard deviation is known and the distribution of the sample mean is approximately normal.

When you perform a hypothesis test of a single population mean [latex]\mu[/latex] using a normal distribution (often called a [latex]z[/latex]-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 doing a hypothesis test for a mean, a [latex]t[/latex]-distribution is used when the sample size is small or you are using the sample standard deviation. This is called a [latex]t[/latex]-test for a mean.
• When doing a hypothesis test for a mean, a normal distribution is used when the sample size is large and the population standard deviation is known. This is called a [latex]z[/latex]-test for a mean.

[latex]z[/latex]-statistic

[latex]z=\dfrac{\stackrel{¯}{x}-μ}{\frac{σ}{\sqrt{n}}}[/latex]

where [latex]\stackrel{¯}{x}[/latex] is the sample (observed) mean, [latex]\mu[/latex] is the population mean, [latex]\sigma[/latex] is the population standard deviation, and [latex]n[/latex] is the sample size.

[latex]t[/latex]-statistic

[latex]t=\dfrac{\stackrel{¯}{x}-μ}{\frac{s}{\sqrt{n}}}[/latex]

where [latex]\stackrel{¯}{x}[/latex] is the sample (observed) mean, [latex]\mu[/latex] is the mean of [latex]\stackrel{¯}{x}[/latex]‘s, [latex]s[/latex] is the standard error of [latex]\stackrel{¯}{x}[/latex], and [latex]n[/latex] is the sample size.

A group of engineers developed a new design for a steel cable. They need to estimate the amount of weight the cable can hold. The weight limit will be reported on the cable packaging.

The engineers take a random sample of [latex]45[/latex] cables and apply weights to each of them until they break. The mean breaking weight for the [latex]45[/latex] cables is [latex]768.2[/latex] lb. and the standard deviation of the breaking weight for the sample is [latex]15.1[/latex] lb. Should we use the [latex]z[/latex]-statistic or the [latex]t[/latex]-statistic as the test statistic to conduct a hypothesis test?

Show answer

Answer: [latex]t[/latex]-statistic

Since we do not know the standard deviation of breaking weights of all of the cables (the population parameter [latex]σ[/latex]), we use the sample standard deviation ([latex]s[/latex]) as an approximation for [latex]σ[/latex]. Since we don’t know [latex]σ[/latex], we must use the [latex]t[/latex]-distribution to model the sampling distribution of means.

Is the [latex]t[/latex]-model a good fit for the sampling distribution?

Yes, because the conditions are met:

• [latex]σ[/latex] is unknown.
• The sample size is large enough.