The high temperatures (in degrees Fahrenheit) of a random sample of 12 small towns are:

Question

The high temperatures (in degrees Fahrenheit) of a random sample of 12 small towns are:

99.3	97.6
98	96.9
97.2	99.5
96.4	96.5
97.8	98.5
98.3	99.1

Assume high temperatures are normally distributed. Based on this data, find the 90% confidence interval of the mean high temperature of towns. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place).

Answer 1

Hi Emma,

We were not given a population standard deviation here, so we have to use a t-confidence interval. Formula for those is:

CI = xbar +/- t*SE, SE= standard error = s/sqrt(n)

Use a TI-80s series calculator or statistical software to compute the mean and standard deviation:

xbar = sample mean = (99.3 +98 + 97.2 + ...)/12

s= sample standard deviation = (x1-xbar)² + (x2-xbar)² + .../ (n-1)

Use software or calculator for at least that second part, possibly both parts.

xbar = 97.93

s = 1.06

Now, for a confidence interval, you need degrees of freedom:

df = n -1, n = sample size

df= 12 - 1 = 11

Go to a t-table online or use software to look up degrees of freedom and 90% confidence. This gives the t-critical value

t* = 1.796

Now, we have xbar and t* but we still need standard error:

SE = s/sqrt(n)

s= 1.06 (above)

n= 12

SE= 1.06/sqrt(12)

SE = 0.31

Now, you can put it all together:

CI = 97.93 +/- (1.796*0.31)

CI = (97.37, 98.49)

I hope this helps.