High temperatures in 10 small towns

Question

The high temperatures (in degrees Fahrenheit) of a random sample of 10 small towns are 97.3, 98.4, 98.5, 96.6, 96.9, 98.9, 98, 97.6, 99, 97.2. Assume high temperatures are normally distributed. Based on this data, find the 98% 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). The answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

Answer 1

Hi Bailey,

We do not know population standard deviation, so this will have to be a t confidence interval. Formula for that is:

CI=x-bar +/- t*SE

Let's break this down:

CI=confidence interval we are looking for

x-bar=sample mean

t*=t critical value

SE=standard error

Right now, we have none of those values, so let's just go one by one:

x-bar=sample mean

For this, we need to add all temperatures and divide by 10.

x-bar=(97.3, 98.4, 98.5, 96.6, 96.9, 98.9, 98, 97.6, 99, 97.2)/10

x-bar=97.84

Now, let's address t*, the t-critical value. We first need to compute degrees of freedom. Formula is:

df=n-1

n=sample size

Thus:

df=10-1

df=9

Now, we go to the t-table, look up 9 degrees of freedom in the row and 98% confidence in the column. This gives:

t*=2.821

Now, we need standard error. To compute this, we take:

SE=s/sqrt(n)

s=sample standard deviation

n=sample size

We were not given sample standard deviation here. The formula for that is:

s=sqrt(sum(x_i-x-bar)²/n)

x_i=individual measurement x₁=97.3, x₂=98.4, etc.

x-bar=sample mean

n=sample size

However, we do not need to do this calculation manually. We can use a calculator or statistical software. From calculator:

s=0.845

But we still don't have our standard error! To get that, recall:

SE=s/sqrt(n)

s=sample standard deviation

n=sample size

s=0.845

n=10

SE=0.845/sqrt(10)

SE=0.267

We finally have everything we need to compute the confidence interval:

CI=x-bar +/- t*SE

CI= 97.84 +/- (2.821*0.267)

CI=97.84 +/- 0.753)

CI=(97.087, 98.593)

I hope this helps.