In: Statistics and Probability
From Barbie dolls to runway models, women in Western countries are exposed to unrealistically thin and arguably unhealthy body standards for their gender. A body mass index (BMI) between 18.5 and 24.9 is considered healthy. Using a 3D computer avatar, participants built what they considered the ideal body of an adult of their own gender. Below appears a summary of the results for a sample of 40 female heterosexual Caucasian undergraduate students from British universities that had been recruited by the researchers. Mean BMI was recorded to be 18.85 with standard deviation 1.75. Does the data provide evidence that young Caucasian women in British Universities, on average, aim for an unhealthy ideal body type (corresponding to a BMI less than 18.5)? Use α = 0.10.
(a) Construct a 99% confidence interval for the mean BMI for young Caucasian women in British universities using R. Use at least 2 decimals in your answer. [2 marks]
(b) Would a 99% CI (for the mean BMI) constructed using a smaller sample than the one used for part a) tend to be wider or narrower? Explain. [2 marks]
(a)
R code:
m=18.85
s=1.75
alpha=0.01
n=40
LL=round(m-qt(1-alpha/2,n-1)*s/sqrt(n),4)#lower limit
UL=round(m+qt(1-alpha/2,n-1)*s/sqrt(n),4)#Upper limit
LL
UL
Output:
> LL
[1] 18.1007
> UL
[1] 19.5993
99% confidence interval for the mean BMI for young Caucasian women in British universities: (18.1007, 19.5993).
b. If sample size is small then Margin of error increases hence length of confidence interval increases hence 99% CI (for the mean BMI) is constructed using a smaller sample than the one used for part a) tend to be wider.
Example:
Take n=30 then 99% CI: (17.9693, 19.7307).
R code:
m=18.85
s=1.75
alpha=0.01
n=30
LL=round(m-qt(1-alpha/2,n-1)*s/sqrt(n),4)#lower limit
UL=round(m+qt(1-alpha/2,n-1)*s/sqrt(n),4)#Upper limit
LL
UL