Confidence Interval & p-value

Confidence Interval

A sample of 50 college students were asked how many exclusive relationships they’ve been in so far. The students in the sample had an average of 3.2 exclusive relationships, with a standard deviation of 1.74. In addition, the sample distribution was only slightly skewed to the right. Estimate the true average number of exclusive relationships based on this sample using a 95% confidence interval.

n=50
μ=3.2
sd=1.74
CI=.95

SE=sd/sqrt(n)
Z=round(qnorm(CI+(1-CI)/2),2)
estimate<-μ+c(-1,1)*Z*SE
print(round(estimate,1))

## [1] 2.7 3.7

p-value

Earlier we calculated a 95% confidence interval for the average number of exclusive relationships college students have been in to be (2.7, 3.7). Based on this confidence interval, do these data support the hypothesis that college students on average have been in more than 3 exclusive relationships.

Ho: μ=3

Ha: μ>3

n=50
μ=3.2
sd=1.74
CI=.95

SE=sd/sqrt(n)
Z=(μ-3)/SE
pvalue<-1-pnorm(Z)
print(round(pvalue,3))

## [1] 0.208

#Since pvalue is high, we do not reject Ho
#pvalue ≈ 0.21, bigger than .05

Interpreting the p-value

• If in fact college students have been in 3 exclusive relationships on average, there is a 21% chance that a random sample of 50 college students would yield a sample mean of 3.2 or higher.
• This is a pretty high probability, so we think that a sample mean of 3.2 or more exclusive relationships is likely to happen simply by chance.

Making a Decision

• Since p-value is high (higher than 5%) we fail to reject Ho.
• The data do not provide convincing evidence that college students have been in more than 3 relationships on average.
• The difference between the null value of 3 relationships and the observed sample mean of 3.2 relationships is due to chance or sampling variability.