Question

This is for Confidence Intervals- Any help?

There is a general consensus that a person’s height is about the same as their wingspan. That is to say, if you calculate “Ape Index” (wingspan minus height in inches) we should get zero on average for a whole population of people. Since Janee has super short arms (ape index of -3) and all her “friends” “lovingly” call her “t-rex arms” Janee would like to think that she’s not alone in the t-rex department and that actually the average Ape Index is a negative number for the general population and that in fact it is her “friend” and coworker Mike Jones who is the super weirdo with his long “sloth” arms and his ape index of +3. Over the last several semesters, she has had her statistics students calculate their Ape index, here are their results: 366 students with an average ape index of -1.1850 inches and a sample standard deviation of 2.8016 inches.

1. Use the sample data to give a 95% confidence interval to estimate the Average Ape Index. Identify what software you are using to give the interval.
2. Clearly describe what the range of values from your confidence interval represents in plain language.
3. What would you say to someone who, like Janee, has a -3 ape index? How about to someone like Mike with a +3 ape index?

Answer 1

1)

Answer:

$\text{95\% CI}=(0.8970,1.4730)$

Explanation:

The confidence interval for the mean is obtained using the formula,

$\text{Confidence Interval}=\overline{x}\pm t_c\times \frac{s}{\sqrt{n}}$

The t critical value is obtained from in excel for the significance level = 0.05 and degree of freedom = n -1 = 366 - 1 = 365.

$t_c=1.966$

now,

$\text{95\% CI}=1.1850\pm 1.9665\times \frac{2.8016}{\sqrt{366}}$

$\text{95\% CI}=(0.8970,1.4730)$

The calculations are done in excel. The screenshot is shown below,

B7 x fc =T.INV.2T(B6,B5) A D 1 2 Sample mean 3 Sample std dev 4 Sample size 5 df 6 alpha 7 t critical 8 9 lower limit 10 upper limit 11 B с Excel function 1.1850 2.8016 366 365 D4-1 0.05 1.9665.T.INV.2T(D6,05) 0.8970 D2-D7*D3/SQRT(D4) 1.4730 D2+D7*D3/SQRT(D4)

2)

Answer:

We are 95% confident that the mean Ape Index (wingspan minus height) for the people lies between 0.8970 and 1.4730 inches.

Explanation:

The 95% confidence interval for the mean Ape Index is (0.9870,1.4730) which means we are 95% confident that the mean Ape Index (wingspan minus height) for the people lies between 0.8970 and 1.4730 inches.

3)

For Janee,

The Ape Index is -3 which is less than the lower limit = 0.8750 of the 95% confidence interval hence we can conclude that the Janee's Ape Index is significantly less than the mean ape index of people at a 2.5% significance level.

For Mike

The Ape Index is 3 which is greater than the upper limit = 1.4730 of the 95% confidence interval hence we can conclude that Mike's Ape Index is significantly higher than the mean ape index of people at 2.5% significance level.