A random sample of 64 door-to-door encyclopedia salespersons were asked how long on average they were able to talk to the potential customer. Their answers revealed a mean of 8.5 minutes. The population standard deviation is 3 minutes.
Construct a 95% confidence interval for mu, the time it takes an encyclopedia salesperson to talk to a potential customer.
What is the upper confidence limit?
Answer:
The 95% confidence interval for mu, the time it takes an encyclopedia salesperson to talk to a potential customer is between 7.765 minutes and 9.235 minutes.
The upper confidence limit is of 9.12 minutes.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1 – 0.95}{2} = 0.025[/tex]
Now, we have to find z in the Z-table as such z has a p-value of [tex]1 – \alpha[/tex].
That is z with a pvalue of [tex]1 – 0.025 = 0.975[/tex], so Z = 1.96.
Now, find the margin of error M as such
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
[tex]M = 1.96\frac{3}{\sqrt{64}} = 0.735[/tex]
The lower end of the interval is the sample mean subtracted by M. So it is 8.5 – 0.735 = 7.765 minutes
The upper end of the interval is the sample mean added to M. So it is 8.5 + 0.735 = 9.235 minutes
The 95% confidence interval for mu, the time it takes an encyclopedia salesperson to talk to a potential customer is between 7.765 minutes and 9.235 minutes.
What is the upper confidence limit?
Similar procedue above, just a few changes.
Now Z with a p-value of 0.95, so Z = 1.645.
[tex]M = 1.645\frac{3}{\sqrt{64}} = 0.62[/tex]
8.5 + 0.62 = 9.12 minutes
The upper confidence limit is of 9.12 minutes.