To calculate a 95% confidence interval for the mean difference between measured and believed heights, we follow these steps:

Step 1: Define the differences

Calculate the differences $d = \text{measured} - \text{believed}$ for each pair.

Step 2: Compute statistics for the differences

1. Find the mean difference ($\bar{d}$).
2. Find the standard deviation of the differences ($s_d$).
3. Calculate the standard error of the mean ($SE = \frac{s_d}{\sqrt{n}}$).

Step 3: Determine the t-value

Find the critical $t$-value for $n - 1$ degrees of freedom at a 95% confidence level.

Step 4: Compute the confidence interval

The confidence interval is given by: $\bar{d} \pm t \cdot SE$

Perform the calculations:

I will calculate this for you.import numpy as np from scipy.stats import t

Data

measured = np.array([165, 174, 182, 164, 191, 172, 176, 190, 180, 179, 181, 175, 173, 172, 184]) believed = np.array([167.64, 172.72, 187.96, 167.64, 195.58, 175.26, 177.80, 193.04, 180.34, 177.80, 182.88, 172.72, 167.64, 167.64, 180.34])

Calculate differences

differences = measured - believed

Mean and standard deviation of differences

mean_diff = np.mean(differences) std_diff = np.std(differences, ddof=1) # sample standard deviation n = len(differences)

Standard error of the mean

SE = std_diff / np.sqrt(n)

Degrees of freedom

df = n - 1

t critical value for 95% confidence

t_critical = t.ppf(0.975, df)

Confidence interval

lower_bound = mean_diff - t_critical * SE upper_bound = mean_diff + t_critical * SE

lower_bound, upper_bound(-2.5154300935302305, 1.3154300935302343)(-2.5154300935302305, 1.3154300935302343)successThe 95% confidence interval for the mean difference (measured - believed) is approximately:

$(-2.515, 1.315)$

Interpretation:

• The interval captures $0$, indicating that there is no statistically significant difference between the measured and believed heights at the 95% confidence level. This suggests that the discrepancies between measured and believed heights could be due to random variation rather than a systematic difference.

Would you like a detailed explanation of any of these steps?

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5 Related Questions:

1. How do you compute a confidence interval for paired data?
2. What is the importance of checking the assumptions for a $t$-test?
3. Can a confidence interval that captures $0$ still be practically meaningful?
4. How does the sample size affect the width of a confidence interval?
5. What are alternative methods to analyze paired data?

Tip: Always verify the assumptions of normality and independence before conducting a $t$-test for paired samples.