Lesson 12: One Mean T-Test

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Day 1

Day 2

Exploration Exercise 1.5

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Date Start End
Wed, 17 Dec 2025 1300 1630
Thu, 18 Dec 2025 0730 1100

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12/120

10:37

Bunny

One Mean T-Test

Review: \(z\)-Tests for One Proportion

For all cases:

\[ z = \frac{\hat{p} - \pi_0}{\sqrt{\frac{\pi_0 (1 - \pi_0)}{n}}} \]

Alternative Hypothesis Formula for \(p\)-value R Code
\(H_A: p > \pi_0\) \(p = 1 - \Phi(z)\) p_val <- 1 - pnorm(z_stat)
\(H_A: p < \pi_0\) \(p = \Phi(z)\) p_val <- pnorm(z_stat)
\(H_A: p \neq \pi_0\) \(p = 2 \cdot (1 - \Phi(|z|))\) p_val <- 2 * (1 - pnorm(abs(z_stat)))

Where:

  • \(\hat{p} = R/n\) (sample proportion)
  • \(\pi_0\) = hypothesized proportion under \(H_0\)
  • \(\Phi(\cdot)\) = cumulative distribution function (CDF) of the standard normal distribution.

One Mean T-Test (Less than alternative)

We want to test whether Male cadets (Don’t worry ladies, we’ll get to you) are shorter than the average height of U.S. men aged 19–24, which is 72 inches.

Hypotheses

\[ H_0 : \mu = 72 \]

\[ H_A : \mu < 72 \]

Sample Data

heights <- c(70, 71, 69, 73, 68, 74, 71, 70, 72, 69, 70, 71, 68, 73)
WarningIs This a Good Sample?

We measured the heights of 14 cadets in this class.
But our research question is about all U.S. men aged 19–24.

  • Do cadets in this class represent the broader population (what is that population)?
  • What kinds of biases might be present?
  • If the sample isn’t random, how should that affect our conclusions?

Test Statistic Formula

\[ t \;=\; \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \qquad df = n - 1 \]

\[ \begin{aligned} \bar{x} & = \text{sample mean} \\[4pt] \mu_0 & = \text{hypothesized mean (72)} \\[4pt] s & = \text{sample standard deviation} \\[4pt] n & = \text{sample size} \\[4pt] t & = \text{test statistic} \end{aligned} \]

Sample Statistics

n     <- length(heights)
xbar  <- mean(heights)
s     <- sd(heights)

c(n = n, mean = round(xbar, 2), sd = round(s, 2))
    n  mean    sd 
14.00 70.64  1.86

Calculate the Test Statistic

mu0 <- 72
t_stat <- (xbar - mu0) / (s / sqrt(n))
t_stat
[1] -2.722848

Visualize the t Distribution

df <- n - 1

ggplot() +
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  labs(title = "t-Distribution with df = 13")

Add the observed test statistic:

ggplot() +
  stat_function(fun = dt, args = list(df = df), xlim = c(-4, t_stat),
                geom = "area", fill = "darkblue", alpha = 0.5) +
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  geom_vline(xintercept = t_stat, color = "firebrick", linewidth = 1.2) +
  labs(title = "Left-tail Shaded: t-Distribution with Test Statistic")

p-value

p_val <- pt(t_stat, df = df)   
p_val
[1] 0.008708986

Conclusion

With p ≈ .008, we do have evidence that male cadets are shorter than 72 inches.

One Mean T-Test (Greater than alternative)

We want to test whether female cadets are taller than the average height of U.S. women aged 19–24, which is 64 inches.

Hypotheses

\[ H_0 : \mu = 64 \]

\[ H_A : \mu > 64 \]

Sample Data

heights <- c(65, 67, 66, 64)

Test Statistic Formula

\[ t \;=\; \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \qquad df = n - 1 \]

Sample Statistics

n     <- length(heights)
xbar  <- mean(heights)
s     <- sd(heights)

c(n = n, mean = round(xbar, 2), sd = round(s, 2))
    n  mean    sd 
 4.00 65.50  1.29

Calculate the Test Statistic

mu0 <- 64
t_stat <- (xbar - mu0) / (s / sqrt(n))
t_stat
[1] 2.32379

Visualize the t Distribution

df <- n - 1

ggplot() +
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  labs(title = "t-Distribution with df = 3")

Add the observed test statistic and shade the right tail:

ggplot() +
  stat_function(fun = dt, args = list(df = df), xlim = c(t_stat, 4),
                geom = "area", fill = "darkred", alpha = 0.5) +
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  geom_vline(xintercept = t_stat, color = "firebrick", linewidth = 1.2) +
  labs(title = "Right-tail Shaded: t-Distribution with Test Statistic")

p-value

p_val <- 1 - pt(t_stat, df = df)   # one-tailed, greater than
p_val
[1] 0.05136404

Conclusion

With p ≈ .051, we do not have strong evidence that female cadets are taller than 64 inches.

One Mean T-Test (Two-Tailed Alternative)

Suppose the national average height of men aged 19–24 is 72 inches.
We want to test whether West Point Department of Math instructors are different (either taller or shorter).

Hypotheses

\[ H_0 : \mu = 72 \]

\[ H_A : \mu \neq 72 \]

Sample Data

Here is a sample of heights (in inches) from 28 Math instructors:

heights <- c(71, 70, 73, 72, 74, 69, 71, 72, 70, 73, 72, 71, 75, 70, 72, 74, 71, 69, 73, 72, 70, 71, 74, 72, 70, 73, 71, 72)

Test Statistic Formula

\[ t \;=\; \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \qquad df = n - 1 \]

Sample Statistics

n     <- length(heights)
xbar  <- mean(heights)
s     <- sd(heights)

c(n = n, mean = round(xbar, 2), sd = round(s, 2))
    n  mean    sd 
28.00 71.68  1.56

Calculate the Test Statistic

mu0 <- 72
t_stat <- (xbar - mu0) / (s / sqrt(n))
t_stat
[1] -1.086979

Visualize the t Distribution

df <- n - 1

ggplot() +
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  labs(title = paste("t-Distribution with df =", df))

Add the observed test statistic and shade both tails:

ggplot() +
  stat_function(fun = dt, args = list(df = df), xlim = c(-4, -abs(t_stat)),
                geom = "area", fill = "steelblue", alpha = 0.5) +
  stat_function(fun = dt, args = list(df = df), xlim = c(abs(t_stat), 4),
                geom = "area", fill = "steelblue", alpha = 0.5) +
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  geom_vline(xintercept = t_stat, color = "firebrick", linewidth = 1.2) +
  geom_vline(xintercept = -t_stat, color = "firebrick", linewidth = 1.2) +
  labs(title = "Two-tailed Shaded: t-Distribution with Test Statistic")

p-value

p_val <- 2 * (1 - pt(abs(t_stat), df = df))   # two-tailed
p_val
[1] 0.286655

Conclusion

With p ≈ .29, we do not have evidence that Math instructors’ heights differ from 72 inches.

T Distribution Exploration

Overlay several \(t\) distributions with different degrees of freedom and the standard normal \(N(0,1)\) for comparison.

Board Problems

Coffee Consumption

Nationally, the average college student drinks about \(3.2\) cups of coffee per day.
You suspect cadets might consume a different amount (not necessarily more or less).

You collect a sample of \(12\) cadets with the following self-reported daily coffee consumption (in cups):

coffee <- c(2.5, 3.0, 3.8, 4.1, 3.2, 2.9, 3.6, 4.0, 3.3, 2.7, 3.5, 3.9)

Tasks:

  1. State the null and alternative hypotheses.
  2. Compute the sample mean \(\bar{x}\), standard deviation \(s\), and sample size \(n\).
  3. Write down the test statistic formula for a one-sample \(t\)-test.
  4. Calculate the test statistic.
  5. Find the \(p\)-value for the appropriate two-tailed test.
  6. State your conclusion in the context of the problem.

Helpful formula

\[ t \;=\; \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, \qquad df = n-1 \]

1. Hypotheses

\[ H_0: \mu = 3.2 \qquad\text{vs}\qquad H_A: \mu \neq 3.2 \]

2. Descriptive statistics (R)

n    <- length(coffee)
xbar <- mean(coffee)
s    <- sd(coffee)
c(n = n, mean = round(xbar, 3), sd = round(s, 3))
     n   mean     sd 
12.000  3.375  0.528

Numerically: \(n = 12\), \(\bar{x} \approx 3.375\), \(s \approx 0.528\).

3–4. Test statistic

mu0   <- 3.2
t_stat <- (xbar - mu0) / (s / sqrt(n))
df     <- n - 1
c(t_stat = round(t_stat, 3), df = df)
t_stat     df 
 1.149 11.000

Numerically: \(t \approx 1.149\) with \(df = 11\).

5. Two-tailed \(p\)-value

p_val <- 2 * (1 - pt(abs(t_stat), df = df))
p_val
[1] 0.2749615

Numerically: \(p \approx 0.275\).

(Optional) Visual: Two-tailed shading for the observed statistic

library(ggplot2)

ggplot() +
  # Left tail shading
  stat_function(fun = dt, args = list(df = df), xlim = c(-4, -abs(t_stat)),
                geom = "area", fill = "darkblue", alpha = 0.5) +
  # Right tail shading
  stat_function(fun = dt, args = list(df = df), xlim = c(abs(t_stat), 4),
                geom = "area", fill = "darkblue", alpha = 0.5) +
  # Overlay t density
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  # Vertical lines at ±t_stat
  geom_vline(xintercept = c(-abs(t_stat), abs(t_stat)),
             color = "firebrick", linewidth = 1.2, linetype = "dashed") +
  labs(title = "Two-tailed Shaded: t-Distribution with Test Statistic",
       x = "t", y = "Density")

6. Conclusion

With \(p \approx 0.275\), we do not have evidence that cadets’ average coffee consumption differs from \(3.2\) cups per day.

Daily Screen Time

A recent campus wellness report suggests the average college student spends \(3.0\) hours per day on recreational screen time (not including coursework).
You suspect students in your section spend more than that.

You collect a sample of \(12\) students with the following daily screen-time values (in hours):

screen_time <- c(2.5, 3.0, 3.1, 3.2, 3.3, 3.4, 3.8, 3.7, 3.8, 3.2, 3.5, 3.5)

Tasks: 1. State the null and alternative hypotheses.
2. Compute the sample mean \(\bar{x}\), standard deviation \(s\), and sample size \(n\).
3. Write down the test statistic formula for a one-sample \(t\)-test.
4. Calculate the test statistic.
5. Find the one-tailed \(p\)-value for the “greater than” test.
6. State your conclusion in context.

1. Hypotheses \[ H_0: \mu = 3.0 \qquad\text{vs}\qquad H_A: \mu > 3.0 \]

2. Descriptive statistics (R)

n    <- length(screen_time)
xbar <- mean(screen_time)
s    <- sd(screen_time)
c(n = n, mean = round(xbar, 3), sd = round(s, 3))
     n   mean     sd 
12.000  3.333  0.373

3–4. Test statistic

mu0    <- 3.0
t_stat <- (xbar - mu0) / (s / sqrt(n))
df     <- n - 1
c(t_stat = round(t_stat, 3), df = df)
t_stat     df 
   3.1   11.0

5. One-tailed \(p\)-value (\(H_A: \mu > \mu_0\))

p_val <- 1 - pt(t_stat, df = df)
p_val
[1] 0.005056453

6. Conclusion

With the computed \(t\) and \(p\) above, interpret whether there is evidence that average daily recreational screen time in this section exceeds \(3.0\) hours.

(Optional) Visual: right-tail shading for the observed statistic

library(ggplot2)

ggplot() +
  stat_function(fun = dt, args = list(df = df), xlim = c(t_stat, 4),
                geom = "area", alpha = 0.5) +
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  geom_vline(xintercept = t_stat, linewidth = 1.2) +
  labs(title = "Right-tail shaded: t-Distribution with Test Statistic",
       x = "t", y = "Density")

Weeknight Sleep

Public health guidelines recommend \(7.0\) hours of sleep on weeknights.
You suspect students in your section average less than that.

You collect a sample of \(15\) students’ self-reported weeknight sleep (in hours):

sleep <- c(6.6, 6.9, 7.1, 6.8, 7.0, 6.7, 6.5, 6.8, 6.9, 6.4, 6.6, 7.2, 6.7, 6.8, 6.5)

Tasks: 1. State the null and alternative hypotheses.
2. Compute the sample mean \(\bar{x}\), standard deviation \(s\), and sample size \(n\).
3. Write down the test statistic formula for a one-sample \(t\)-test.
4. Calculate the test statistic.
5. Find the one-tailed \(p\)-value for the “less than” test.
6. State your conclusion in context.

Helpful formula

\[ t \;=\; \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, \qquad df = n-1 \]

1. Hypotheses \[ H_0: \mu = 7.0 \qquad\text{vs}\qquad H_A: \mu < 7.0 \]

2. Descriptive statistics (R)

n    <- length(sleep)
xbar <- mean(sleep)
s    <- sd(sleep)
c(n = n, mean = round(xbar, 3), sd = round(s, 3))
     n   mean     sd 
15.000  6.767  0.229

3–4. Test statistic

mu0    <- 7.0
t_stat <- (xbar - mu0) / (s / sqrt(n))
df     <- n - 1
c(t_stat = round(t_stat, 3), df = df)
t_stat     df 
-3.949 14.000

5. One-tailed \(p\)-value (\(H_A: \mu < \mu_0\))

p_val <- pt(t_stat, df = df)
p_val
[1] 0.0007279758

6. Conclusion

Report the computed \(t\), \(df\), and \(p\), then conclude whether there is evidence that average weeknight sleep is less than \(7.0\) hours.

(Optional) Visual: Left-tail shading for the observed statistic

library(ggplot2)

ggplot() +
  # Left tail shading up to t_stat
  stat_function(fun = dt, args = list(df = df), xlim = c(-4, t_stat),
                geom = "area", fill = "darkblue", alpha = 0.5) +
  # Overlay t density
  geom_function(fun = dt, args = list(df = df), xlim = c(-4, 4)) +
  # Vertical line at t_stat
  geom_vline(xintercept = t_stat,
             color = "firebrick", linewidth = 1.2, linetype = "dashed") +
  labs(title = "Left-tail Shaded: t-Distribution with Test Statistic",
       x = "t", y = "Density")

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