Question

Suppose you are a researcher in a hospital. You are experimenting with a new tranquilizer. You collect data from a random sample of 9 patients. The period of effectiveness of the tranquilizer for each patient (in hours) is as follows:

2.5
2.8
2
2.1
2.6
2.5
2.6
2.6
2.9

a. What is a point estimate for the population mean length of time. (Round answer to 4 decimal places)

b. Which distribution should you use for this problem?

• normal distribution
• t-distribution

c. Why?


d. What must be true in order to construct a confidence interval in this situation?

• The population must be approximately normal
• The sample size must be greater than 30
• The population mean must be known
• The population standard deviation must be known

e. Construct a 98% confidence interval for the population mean length of time. Enter your answer as an open-interval (i.e., parentheses) Round upper and lower bounds to two decimal places

f. Interpret the confidence interval in a complete sentence. Make sure you include units

g. What does it mean to be "98% confident" in this problem? Use the definition of confidence level.

• There is a 98% chance that the confidence interval contains the population mean
• 98% of all simple random samples of size 9 from this population will result in confidence intervals that contain the population mean
• The confidence interval contains 98% of all samples

h. Suppose that the company releases a statement that the mean time for all patients is 2 hours.

Is this possible?

• Yes
• No

Is it likely?

• No
• Yes

i. Use the results above and make an argument in favor or against the company's statement. Structure your essay as follows:

1. Describe the population and parameter for this situation.
2. Describe the sample and statistic for this situation.
3. Give a brief explanation of what a confidence interval is.
4. Explain what type of confidence interval you can make in this situation and why.
5. Interpret the confidence interval for this situation.
6. Restate the company's claim and whether you agree with it or not.
7. Use the confidence interval to estimate the likelihood of the company's claim being true.
8. Suggest what the company should do.

Answer 1

a. What is a point estimate for the population mean length of time.

Point estimate for population mean = Sample mean = Xbar = 2.5111

b. Which distribution should you use for this problem?

t-distribution

c. Why?

...because we do not given the value for the population standard deviation and sample size is small (n<30).

d. What must be true in order to construct a confidence interval in this situation?

The population must be approximately normal.

e. Construct a 98% confidence interval for the population mean length of time.

Confidence interval for Population mean is given as below:

Confidence interval = Xbar ± t*S/sqrt(n)

From given data, we have

Xbar = 2.511111111

S = 0.293446948

n = 9

df = n – 1 = 9 – 1 = 8

Confidence level = 98%

Critical t value = 2.8965

Confidence interval = Xbar ± t*S/sqrt(n)

Confidence interval = 2.511111111 ± 2.8965*0.293446948/sqrt(9)

Confidence interval = 2.511111111 ± 2.8965*0.097815649

Confidence interval = 2.511111111 ± 0.2833

Lower limit = 2.511111111 - 0.2833 = 2.23

Upper limit = 2.511111111 + 0.2833 = 2.79

Confidence interval = (2.23, 2.79)

f. Interpret the confidence interval in a complete sentence.

We are 98% confident that the population mean length of time will lies between 2.23 hours to 2.79 hours.

g. What does it mean to be "98% confident" in this problem?

98% of all simple random samples of size 9 from this population will result in confidence intervals that contain the population mean

h. Suppose that the company releases a statement that the mean time for all patients is 2 hours.

Is this possible?

No

Is it likely?

No