Math Problem Statement
You are interested in finding a 90% confidence interval for the mean number of visits for physical therapy patients. The data below show the number of visits for 11 randomly selected physical therapy patients. Round answers to 3 decimal places where possible.
10,6,6,6,24,15,27,9,5,27,18 a. To compute the confidence interval use a distribution.
between
b. With 90% confidence the population mean number of visits per physical therapy patient is and visits.
c. If many groups of 11 randomly selected physical therapy patients are studied, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population mean number of visits per patient and about percent will not contain the true population mean number of visits per patient.
Solution
To find a 90% confidence interval for the mean number of visits for physical therapy patients based on the given data, we can follow these steps:
Given Data:
Step 1: Calculate the Sample Mean ()
Step 2: Calculate the Sample Standard Deviation (s)
First, calculate the squared deviations:
Sum of squared deviations: Now, calculate the sample standard deviation:
Step 3: Determine the t-Score for a 90% Confidence Interval
Since the sample size is small (n = 11), we use the t-distribution. Degrees of freedom () is . For a 90% confidence interval, the critical t-value () is approximately 1.812.
Step 4: Calculate the Margin of Error (ME)
Step 5: Compute the Confidence Interval
Answers:
a. To compute the confidence interval, use the t-distribution.
b. With 90% confidence, the population mean number of visits per physical therapy patient is between 9.117 and 18.701 visits.
c. If many groups of 11 randomly selected physical therapy patients are studied, then a different confidence interval would be produced from each group. About 90% of these confidence intervals will contain the true population mean number of visits per patient, and about 10% will not contain the true population mean number of visits per patient.
Do you want more details or have any questions?
Related Questions:
- How do you interpret a 90% confidence interval?
- Why is the t-distribution used instead of the normal distribution in this case?
- What would happen to the confidence interval if the sample size was larger?
- How does the sample standard deviation affect the confidence interval?
- What is the meaning of the margin of error in the context of this problem?
- How would you calculate a 95% confidence interval for the same data?
- What are the assumptions underlying the confidence interval calculation?
- How can confidence intervals be used in decision-making?
Tip:
When working with small sample sizes (n < 30), it's important to use the t-distribution rather than the z-distribution, as it accounts for the additional uncertainty in estimating the population standard deviation.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
T-distribution
Formulas
Sample Mean
Sample Standard Deviation
Margin of Error
Theorems
Central Limit Theorem
Suitable Grade Level
Advanced High School
Related Recommendation
Calculating a 90% Confidence Interval for Physical Therapy Patient Visits
95% Confidence Interval for Physical Therapy Visits Using t-Distribution
95% Confidence Interval for Mean Number of Visits for Physical Therapy Patients
Constructing a 90% Confidence Interval for Population Mean in Psychology Examinations
90% Confidence Interval for Population Mean Using Sample Data