Question

An article investigated the consumption of caffeine among women. A sample of 52 women were asked to monitor their caffeine intake over the course of one day. The mean amount of caffeine consumed in the sample of women was 226.553 mg with a standard deviation of 229.462 mg. In the article, researchers would like to include a 99% confidence interval.   

• The values below are t critical point corresponding to 90%, 95%, 98%, and 99% confidence. Which critical point corresponds to 99% confidence?
  • 2.676
  • 2.008
  • 1.675
  • 2.402
• We are 99% confident that the true mean amount of caffeine consumed by women is between ___ mg and ____ mg. (Round the limits of your interval to 4 decimal places.)
• If the level of confidence were changed to 98%, what effect would this have on the width of the calculated confidence interval?
  • there would be no effect on the interval
  • the interval would widen
  • the interval would narrow

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 226.553

sample standard deviation = s = 229.462

sample size = n = 52

Degrees of freedom = df = n - 1 = 51

At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

t/2,df = t_0.005,51 = 2.676

Margin of error = E = t_/2,df * (s /n)

= 2.676 * (229.462 / 52)

Margin of error = E = 85.1521

The 99% confidence interval estimate of the population mean is,

  ± E  

= 226.553 ± 85.1521

= ( 141.4009, 311.7051 )

We are 99% confident that the true mean amount of caffeine consumed by women is between 141.4009 mg and 311.7051 mg.

The interval would narrow, because confidence level decreases, margin of error decreases.