To find the 95% confidence interval estimate of the true mean red blood cell count ($\mu$), we will use the formula:

$CI = \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}$

Where:

• $\bar{x}$: sample mean
• $t^*$: critical value for the 95% confidence level (depends on degrees of freedom, $df = n-1$)
• $s$: sample standard deviation
• $n$: sample size

Step-by-Step Calculation:

1. Sample Data:
   $5.4, 5.1, 5.0, 5.2, 5.6, 5.3, 5.5, 5.3, 5.2, 5.2, 5.4, 4.8, 5.3, 5.1, 5.3$

2. Calculate Sample Mean ($\bar{x}$):
   $\bar{x} = \frac{\text{Sum of all data points}}{n}$

3. Calculate Sample Standard Deviation ($s$):
   $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

4. Determine Critical Value ($t^*$):
   For a 95% confidence level with $n - 1 = 15 - 1 = 14$ degrees of freedom, use a $t$-table or calculator.

5. Plug Values into the Confidence Interval Formula: $CI = \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}$

Let me calculate these from scipy.stats import t

Data

data = np.array([5.4, 5.1, 5.0, 5.2, 5.6, 5.3, 5.5, 5.3, 5.2, 5.2, 5.4, 4.8, 5.3, 5.1, 5.3])

Sample size

n = len(data)

Sample mean

mean = np.mean(data)

Sample standard deviation

std_dev = np.std(data, ddof=1) # Use ddof=1 for sample standard deviation

Degrees of freedom

df = n - 1

Critical value (t* for 95% confidence interval)

t_critical = t.ppf(1 - 0.025, df)

Margin of error

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

Confidence interval

lower_bound = mean - margin_of_error upper_bound = mean + margin_of_error

mean, std_dev, t_critical, (lower_bound, upper_bound### Results:

• Sample Mean ($\bar{x}$): $5.247$ (rounded to three decimal places)
• Sample Standard Deviation ($s$): $0.200$ (rounded to three decimal places)
• Critical Value ($t^*$): $2.145$ (rounded to three decimal places)

The 95% confidence interval for the true mean red blood cell count is: $(5.136, 5.357)$

Do you have any questions, or would you like more details?

Related Questions:

1. How do we interpret this confidence interval in a biological context?
2. Why do we use the $t$-distribution instead of the normal distribution?
3. How does sample size affect the width of the confidence interval?
4. Can this method be applied to non-normal data distributions?
5. What happens to the confidence interval if we choose a 99% confidence level instead?

Tip:

When calculating confidence intervals, always verify whether the data meets the assumptions