We are conducting a one-sample t-test for paired data (dependent samples) to determine if the mean difference ($\mu_d$) in height between presidents and their main opponents is significantly greater than 0.

Step 1: Compute the Differences ($d$)

The differences between the heights of the presidents and their main opponents are:

$d = \text{Presidents' Heights} - \text{Opponents' Heights}$

Presidents' Heights ($P$)	Opponents' Heights ($O$)	Difference ($d = P - O$)
176	167	9
186	186	0
176	180	-4
186	170	16
193	188	5
170	183	-13

The differences ($d$) are:
$9, 0, -4, 16, 5, -13$

Step 2: Compute Sample Statistics

We compute:

• Mean of differences ($\bar{d}$)
• Standard deviation of differences ($s_d$)
• Sample size ($n = 6$)

Let's calculate these values. numpy as np

Given differences

d = np.array([9, 0, -4, 16, 5, -13])

Compute mean and standard deviation

d_mean = np.mean(d) d_std = np.std(d, ddof=1) # Sample standard deviation (ddof=1 for unbiased estimate) n = len(d)

d_mean, d_std, nThe computed values are:

• Mean difference: $\bar{d} = 2.167$
• Standard deviation: $s_d = 10.187$
• Sample size: $n = 6$

Step 3: Compute the T-Statistic

The t-statistic is given by:

$t = \frac{\bar{d} - 0}{s_d / \sqrt{n}}$

Let's compute it. Compute t-statistic t_stat = d_mean / (d_std / np.sqrt(n)) t_statThe t-statistic is $t = 0.521$.

Step 4: Compute the p-value

Since this is a one-tailed test ($H_1: \mu_d > 0$), we compute:

$p = P(T > t) \text{ for } df = n - 1 = 5$

Let's calculate the p-value. scipy.stats import t

Degrees of freedom

df = n - 1

Compute one-tailed p-value

p_value = 1 - t.cdf(t_stat, df) p_valueThe p-value is $p = 0.312$.

Step 5: Construct the Confidence Interval

The 99% confidence interval for $\mu_d$ is given by:

$\bar{d} \pm t^* \times \frac{s_d}{\sqrt{n}}$

where $t^*$ is the critical value from the t-distribution with $df = 5$ at a 99% confidence level.

Let's compute this interval. Get the critical t-value for 99% confidence level (two-tailed) t_critical = t.ppf(0.995, df) # 0.995 because it's two-tailed

Compute margin of error

margin_error = t_critical * (d_std / np.sqrt(n))

Compute confidence interval

lower_bound = d_mean - margin_error upper_bound = d_mean + margin_error

lower_bound, upper_boundIt seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to explain the method for constructing the confidence interval so you can calculate it manually?