To construct a 99% confidence interval for the mean pulse rate, we follow these steps:

Step 1: Calculate the sample mean ($\bar{x}$) and standard deviation ($s$).

The pulse rates are:
$89, 75, 92, 76, 75, 82, 66, 83, 97, 106, 59, 76, 102$

1. Compute $\bar{x}$ (mean): $\bar{x} = \frac{\text{Sum of all pulse rates}}{\text{Number of pulse rates}}$

2. Compute the sample standard deviation ($s$): $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ where $x_i$ represents each individual data point, and $n$ is the sample size.

Step 2: Find the critical $t$-value.

For a 99% confidence interval with $n - 1 = 12$ degrees of freedom, the critical $t$-value is obtained from a $t$-distribution table.

Step 3: Compute the margin of error (ME):

$ME = t^* \cdot \frac{s}{\sqrt{n}}$

Step 4: Construct the confidence interval:

$\text{Lower Limit} = \bar{x} - ME$ $\text{Upper Limit} = \bar{x} + ME$

Now, I will calculate these values for you.import numpy as np from scipy.stats import t

Given data

pulse_rates = np.array([89, 75, 92, 76, 75, 82, 66, 83, 97, 106, 59, 76, 102]) n = len(pulse_rates)

Step 1: Calculate the sample mean and standard deviation

mean = np.mean(pulse_rates) std_dev = np.std(pulse_rates, ddof=1)

Step 2: Find the critical t-value for 99% confidence interval

confidence_level = 0.99 alpha = 1 - confidence_level degrees_of_freedom = n - 1 t_critical = t.ppf(1 - alpha / 2, degrees_of_freedom)

Step 3: Compute the margin of error

margin_of_error = t_critical * (std_dev / np.sqrt(n))

Step 4: Calculate the confidence interval

lower_limit = mean - margin_of_error upper_limit = mean + margin_of_error

lower_limit, upper_limit(71.20739625916046, 94.63875758699338)(71.20739625916046, 94.63875758699338)successThe 99% confidence interval for the mean pulse rate is:

• Lower limit: 71.2
• Upper limit: 94.6

Would you like a detailed breakdown of the calculations or have any further questions? Here are some related questions to expand on this topic:

1. What is the significance of using a 99% confidence level?
2. How would the interval change if we used a 95% confidence level instead?
3. What is the formula for calculating the t-critical value, and how does it differ from a z-critical value?
4. How would the sample size affect the width of the confidence interval?
5. Can we use this confidence interval to infer the pulse rates of men, or is it limited to women?

Tip: When constructing confidence intervals, always verify whether the sample size is sufficient for the chosen confidence level to ensure reliability.