• Hypothesis Testing is a statistical procedure used to test an assumption about a population using sample data.
• Need to know terms:
  • Null hypothesis (H₀): The null hypothesis is the default assumption that says there is no effect or no difference.
  • Alternative hypothesis (Ha or H₁): The alternative hypothesis is the assumption we are trying to prove.
  • Significance level (α): The significance level is the threshold for determining whether we should or should no reject the null hypothesis. (\(\alpha\) is typically 0.05)
  • P-value: The p-value, or probability, is the value we compare to the significance level.
    • If p ≤ α, we reject H₀ (the result is statistically significant).
    • If p > α, we fail to reject H₀ (not enough evidence against it).

1. State hypotheses (H₀ and Ha).
   
2. Choose Significance level (α).
   
3. Using the appropriate test, calculate the test statistic.
   
4. Find the p-value.
   
5. Conclude and interpret in context.

• The Z-test is a statistical test used to test hypotheses about the population mean or proportions when:
  • The population standard deviation (σ) is known
  • The sample size is large, usually n ≥ 30 (given by the Central Limit Theorem)
• Used to test whether:
  • The sample mean differs from a known population mean
  • Two sample means or proportions differ significantly

• The Central Limit Theorem (CLT) states that if a sample size is large enough, the distribution of sample means will be an approximately normal distribution, regardless of the shape of the original population.
• Conditions that need to be met:
  • Random: Each individual of the population has an equal chance of being selected.
  • Independent: The result of one sample does not affect the outcome of another.
  • Large enough: The sample size must be sufficiently large, typically \(n\ge 30\)

Visual of the Central Limit Theorem

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

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

If |z| > z_critical, reject H₀.

A factory claims that the average weight of its cereal boxes is 500 g. You suspect the boxes might weigh less than advertised.

You take a sample of 40 boxes and find:

• \(\bar{x}\) = 496 g
  
• \(\sigma\) = 10 g
• \(n\) = 40

Hypotheses:

• H0: \(\mu\) = 500
  • The average weight is 500 g
• Ha: \(\mu\) < 500
  • The average weight is less than 500 g

# Sample data
n <- 40
xbar <- 496
mu0 <- 500
sigma <- 10

z = (xbar - mu0) / (sigma / sqrt(n))
z

[1] -2.529822

For a left-tailed test at α = 0.05: \[ p = P(Z \le z) < 0.05 \Rightarrow \text{Reject } H_0 \] Critical z = -1.645

Z = −2.53 < −1.645

p = 0.0057 < 0.05

Conclusion: There is significant evidence at the 5% level that the average weight of cereal boxes is less than 500 g.