Confidence Interval for Two Independent Means

Confidence Interval for Two Means Containing Zero

Comparing two means using confidence intervals for the difference $\mu_1 -\mu_2$ is similar to a hypothesis test of no difference:

$H_o:\mu_1-\mu_2=0$ (no difference)

$H_a:\mu_1-\mu_2\ne0$ (difference)

Example #1

95% confidence interval to estimate $\mu_1 -\mu_2$ is $[36, 70]$

• The confidence interval does not contain zero.
• The LCL and UCL have the same sign (both positive).
• There is evidence of a difference.
• Specifically, $\mu_1$ is greater than $\mu_2$.

Example #2

95% confidence interval to estimate $\mu_1 -\mu_2$ is $[-70, -36]$

• The confidence interval does not contain zero.
• The LCL and UCL have the same sign (both negative).
• There is evidence of a difference.
• Specifically, $\mu_1$ is less than than $\mu_2$.

Example #3

95% confidence interval to estimate $\mu_1 -\mu_2$ is $[-15, 45]$

• The confidence interval contains zero.
• The LCL is negative and UCL is positive.
• There is no evidence of a difference.
• It's possible that $\mu_1$ is greater than $\mu_2$.
• It's also possible that $\mu_1$ is less than $\mu_2$.

Confidence Interval for Two Means (Independent Populations)

We make inferences with confidence intervals.

Example

Ben, the owner of Fitness Planet wonders, if the amount of time spent at the gym differs by gender. His results:

Ben assumes that the two populations have unequal variances. He generates this output:

What is the $p$-value? What do you conclude if you were to do a hypothesis test?

$H_o:\mu_1-\mu_2=0$

$H_a:\mu_1-\mu_2\ne0$ (two-tail test)

The $p$-value is 0.00263. We have very strong evidence that the amount time spent at the gym differs by gender.

Construct a 95% confidence interval. What does it tell you (in plain English)?

We use:

$(\overline{x}_1-\overline{x}_2)\pm t^*\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}$

$\boxed{\begin{array}{l} \begin{array}{ll} df & =\text{the minimum of: } (n_1-1,n_2-1)\\ &=\text{the minimum of: }(84,67)\\ &=67 \end{array}\\ \text{Thus, $t^*$ = 1.996 (using software)} \end{array}}$

$(63-56)\pm1.996\sqrt{\dfrac{(19)^2}{85}+\dfrac{(8)^2}{68}}$

$7\pm4.55$

$[2.45,11.55]$

We are 95% confident that the difference of time spent at the gym is between 2.45 minutes and 11.55 minutes.

Does the interval contain "0"? Interpret that that means.

The interval does not contain "0". This means that one group (specifically the men) is consistently different (specifically greater) than the other group (specifically the women) in terms of time spent at the gym.

Thus, we can conclude that the amount time spent at the gym differs by gender.

$\to$ This is the same conclusion as the hypothesis test.