- Construct 90% and 99% confidence intervals for the population mean temperature of coffee sold in Example 2.
a. Find tc and E for each level of confidence.
b. Use x and E to find the left and right endpoints of each confidence interval.
c. Interpret the results.
- Constructing a Confidence Interval You randomly select 36 cars of the same model that were sold at a
car dealership and determine the number of days each car sat on the dealership’s lot before it was sold. The sample mean is 9.75 days, with a sample standard deviation of 2.39 days. Construct a 99% confidence interval for the population mean number of days the car model sits on the dealership’s lot.
The Correct Answer and Explanation is :
Part 1: Confidence Intervals for the Population Mean Temperature of Coffee
a. Find ( t_c ) and ( E ) for each level of confidence
To construct a confidence interval for the population mean temperature, we use the formula for a confidence interval when the population standard deviation is unknown:
[
CI = \bar{x} \pm t_c \times \frac{s}{\sqrt{n}}
]
Where:
- ( \bar{x} ) is the sample mean,
- ( t_c ) is the critical value from the ( t )-distribution,
- ( s ) is the sample standard deviation,
- ( n ) is the sample size.
We’ll calculate the margin of error ( E = t_c \times \frac{s}{\sqrt{n}} ) for the 90% and 99% confidence levels.
The critical values (( t_c )) are determined by the sample size (( n )) and the degrees of freedom (( df = n – 1 )).
For a 90% confidence interval with 10 degrees of freedom:
- From the ( t )-distribution table, ( t_c = 1.812 ).
For a 99% confidence interval with 10 degrees of freedom:
- From the ( t )-distribution table, ( t_c = 3.169 ).
Now, we compute the margin of error ( E ):
[
E = t_c \times \frac{s}{\sqrt{n}}
]
b. Use ( \bar{x} ) and ( E ) to find the left and right endpoints of each confidence interval
The endpoints of the confidence interval are calculated as:
[
\text{Left endpoint} = \bar{x} – E
]
[
\text{Right endpoint} = \bar{x} + E
]
c. Interpretation of Results
The interpretation of the confidence interval is that we are 90% (or 99%) confident that the true population mean temperature of the coffee falls within the interval. This means that if we repeated this process many times, 90% (or 99%) of the intervals we compute would contain the true population mean temperature.
Part 2: Confidence Interval for the Number of Days Cars Sit on the Lot
Given:
- Sample mean (( \bar{x} )) = 9.75 days,
- Sample standard deviation (( s )) = 2.39 days,
- Sample size (( n )) = 36,
- Confidence level = 99%.
The formula for the confidence interval is the same as in Part 1:
[
CI = \bar{x} \pm t_c \times \frac{s}{\sqrt{n}}
]
First, calculate the standard error of the mean:
[
\text{SE} = \frac{s}{\sqrt{n}} = \frac{2.39}{\sqrt{36}} = 0.398
]
Next, we need the critical value ( t_c ) for 99% confidence. The degrees of freedom (( df = n – 1 )) is 35. From the ( t )-distribution table for 35 degrees of freedom and a 99% confidence level, the critical value ( t_c = 2.724 ).
Now, compute the margin of error ( E ):
[
E = t_c \times \text{SE} = 2.724 \times 0.398 = 1.085
]
So the confidence interval is:
[
CI = 9.75 \pm 1.085
]
This results in the following endpoints:
- Left endpoint: ( 9.75 – 1.085 = 8.665 ),
- Right endpoint: ( 9.75 + 1.085 = 10.835 ).
Interpretation:
We are 99% confident that the true population mean number of days a car sits on the dealership’s lot is between 8.665 and 10.835 days. This means that, based on the sample data, we can expect that most cars of this model will sit on the lot for this range of days before being sold, with a high level of confidence.