The following data gives the amount of time (in minutes), it took a sample of 16 patients with hypertension to achieve their target heart rate during a zumba exercise dance class.
5, 5, 7, 7, 8, 9, 10, 10, 10, 12, 12, 12, 12, 13, 14, 14
Use this data to construct a 95% confidence interval for the true mean amount of time for patients with hypertension to achieve their target heart rate. Along with the sample mean and sample standard deviation, state the confidence interval and a conclusion.
To construct a 95% confidence interval for the true mean amount of time for patients with hypertension to achieve their target heart rate, we can follow these steps:
Step 1: Calculate the sample mean
The sample mean (x̄) can be found by summing up all the data points and dividing by the number of data points (in this case, 16).
x̄ = (5 + 5 + 7 + 7 + 8 + 9 + 10 + 10 + 10 + 12 + 12 + 12 + 12 + 13 + 14 + 14) / 16
Step 2: Calculate the sample standard deviation
The sample standard deviation (s) measures the spread or variability of the data points.
To calculate s, you can use the following formula:
s = √[∑ (x - x̄)² / (n - 1)]
where x is each data point, x̄ is the sample mean, and n is the number of data points.
Step 3: Determine the critical value
The critical value corresponds to the desired level of confidence. For a 95% confidence interval, the critical value can be found using a table or a calculator. The critical value for a 95% confidence level is approximately 2.131.
Step 4: Calculate the margin of error
The margin of error is determined by multiplying the critical value by the standard deviation divided by the square root of the sample size.
Margin of Error = Critical Value * (s / √n)
Step 5: Calculate the confidence interval
The confidence interval is calculated by subtracting the margin of error from the sample mean and adding it to the sample mean.
Confidence Interval = Sample Mean ± Margin of Error
Now, let's calculate these values using the given data:
Step 1: Sample Mean:
x̄ = (5 + 5 + 7 + 7 + 8 + 9 + 10 + 10 + 10 + 12 + 12 + 12 + 12 + 13 + 14 + 14) / 16
x̄ = 10
Step 2: Sample Standard Deviation:
s = √[∑ (x - x̄)² / (n - 1)]
s = √[ (5 - 10)² + (5 - 10)² + (7 - 10)² + ... + (14 - 10)² ] / (16 - 1)
s ≈ 2.856
Step 3: Critical Value (for 95% confidence interval):
The critical value for a 95% confidence interval is approximately 2.131.
Step 4: Margin of Error:
Margin of Error = Critical Value * (s / √n)
Margin of Error = 2.131 * (2.856 / √16)
Margin of Error ≈ 1.036
Step 5: Confidence Interval:
Confidence Interval = Sample Mean ± Margin of Error
Confidence Interval = 10 ± 1.036
Confidence Interval ≈ (8.964, 11.036)
Conclusion:
Based on the given data, we can conclude that we are 95% confident that the true mean amount of time for patients with hypertension to achieve their target heart rate falls within the interval (8.964, 11.036) minutes.