6.5 Confidence Intervals for a Mean

• For samples of size $n \ge 30$, $\bar{X}$ will be approximately normal even if $X$ isn’t.

• In this case, we can plug in $s$ for $\sigma$: $$\bar{x} \pm z_{\alpha/2}\frac{s}{\sqrt{n}}.$$

In 2010, a survey collected responses from 1,154 US residents. One of the questions was “After an average workday, about how many hours do you have to relax or pursue activities that you enjoy?”. The average time spent relaxing was 3.68 hours, with a standard deviation of 2.6 hours.

Find and interpret a 95% confidence interval for the average time spent relaxing after work.

What do we do when $n < 30$?

If $$Z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}$$ has a standard normal distribution (for $X$ normal or $n\ge30$), the slightly modified $$T = \frac{\bar{X}-\mu}{s/\sqrt{n}}$$ has what we call the t-distribution with $n-1$ degrees of freedom.

The only thing we need to know about degrees of freedom is that $\text{df}=n-1$ is the t-distribution’s only parameter.

• The t-distribution is symmetric and always centered at 0.
• When $n\ge30$, the t-distribution is approximately equivalent to the standard normal distribution.
• For smaller sample sizes, the t-distribution has more area in the tails.

• Plug in $s$ for $\sigma$ and use a t critical value $t_{\text{df}, \, \alpha/2}$.
  • The t critical value is the $[1- \alpha/2]$th percentile of the t-distribution with $n-1$ degrees of freedom.
• The resulting 95% confidence interval is $$\bar{x} \pm t_{\text{df}, \, \alpha/2}\frac{s}{\sqrt{n}}.$$

Rossman and Chance t Probability Calculator

• For this applet, enter the degrees of freedom $n-1$ next to “df”.
• Then check the top box under “t-value probability” and make sure the inequality is clicked to “>” .
• Enter the value of $\alpha/2$ for the probability.
• Click anywhere else on the page and the applet will automatically fill in the box under “t-value”.
• This is your t critical value.

Find t critical values for the following settings:

1. A $95\%$ confidence interval with $n=17$
2. A $90\%$ confidence interval with $n=8$
3. A $98\%$ confidence interval with $n=24$

The weights of 6 randomly selected NFL linebackers were $$243, 238, 229, 253, 248, 225$$ . Find and interpret a 90% confidence interval for the average weight of NFL linebackers.

A different sample of NFL linebacker weights from 23 randomly selected players resulted in a mean of 242 pounds and a standard deviation of 13 pounds. Based on this new sample, find and interpret a 98% confidence interval for the average weight of NFL linebackers.

Section 6.5 Exercises 1-4