• Hypothesis testing is a method of making decisions using data.
• We test an assumption (the “null hypothesis”) against an alternative.
• Common in science, medicine, and engineering.

• Null Hypothesis (\(H_0\)): Status quo or default claim.
• Alternative Hypothesis (\(H_1\) or \(H_a\)): What we hope to support.
• Decision is based on sample data — never 100% certainty.

• A battery manufacturer claims their batteries last at least 100 hours on average.

• From production data, the manufacturer calculated the population standard deviation to be 5

• Another competitor tests 50 batteries and gets a mean of 98.5 hours.

• Is this enough evidence to reject the manufacturer’s claim?

We use a one-sample z-test for the mean to compare a sample mean (obtained from the competitor) to a known population mean (as claimed by the manufacturer):

\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

Where:
- \(\bar{x}\) = sample mean
- \(\mu_0\) = claimed population mean
- \(\sigma\) = population standard deviation
- \(n\) = sample size

• This plot displays the standard normal distribution (mean = 0, SD = 1), which represents the null hypothesis \(H_0\).
• The black curve shows the shape of the z-distribution under \(H_0\).
• The red line at \(z = -2.12\) marks our observed test statistic, calculated from the sample data.
• This z-score tells us how many standard errors the observed sample mean (98.5) is below the hypothesized population mean (100).

• Here we simulated 1000 random samples of size 50.
• Each sample is drawn from a population with true mean 100 (manufacturer claim) and standard deviation of 5.
• The plot shows the distribution of the sample means.
• The observed sample mean (98.5) is far into the tail of the sampling distribution under H₀ (centered at 100), making it unlikely to occur by chance — suggesting statistical significance.

• The p-value is the probability of getting a test statistic at least as extreme as ours, under \(H_0\).
• If \(p < α\) (e.g., 0.05), we reject \(H_0\).
• Lower p-values indicate stronger evidence against \(H_0\).

z_test <- function(xbar, mu0, sigma, n) {
  z <- (xbar - mu0) / (sigma / sqrt(n))
  p <- 2 * (1 - pnorm(abs(z)))
  list(z = z, p = p)
}

z_test(98.5, 100, 5, 50)

## $z
## [1] -2.12132
## 
## $p
## [1] 0.03389485

• The z-test statistic falls in the shaded rejection region beyond ±1.96 (from α = 0.05 according to standard practice), indicating a significant result.

• Since the visualized region p-value (red) extends outside the blue line (p < α with 0.034 < 0.05) we reject the null hypothesis (the base claim by the manufacturer)