Hypothesis Testing Exercises on Height

Hypothesis Testing for Height

• Goal: Show that the mean height of a population is greater than 70 inches.

• Initial Parameters:

  • Sample size: 50 individuals
  • Population standard deviation ($\sigma$): 3 inches
  • Significance level ($\alpha$): 0.05

• Hypotheses:

  • Null Hypothesis ($H_0$): Mean height $\leq$ 70 inches
  • Alternative Hypothesis ($H_a$): Mean height $>$ 70 inches (claim to disprove)

• Statistical Method:

  • Statistic: Sample mean height of 50 people
  • Rejection Rule: If $z<em>{\bar{x}} > z</em>{\alpha}$, reject $H_0$.
  • Step 1: Check if Samping Distribution is Normal.

• Decision-Making Process:

  • Collect a sample of height data and calculate the sample mean.
  • Calculate the $z$-score using the formula:
    $z = \frac{\bar{x} - 70}{\sigma / \sqrt{n}}$
  • Compare calculated $z$-score with $z<em>{\alpha}$. If calculated $z$ exceeds $z</em>{\alpha}$, reject null hypothesis.

Alternative Hypothesis Testing for Equality

• Goal: Show that the mean height is equal to 70 inches.

• Initial Parameters:

  • Sample size: 30 individuals
  • Population standard deviation ($\sigma$): 3 inches
  • Significance level ($\alpha$): 0.05

• Hypotheses:

  • Null Hypothesis ($H_0$): Mean height = 70 inches
  • Alternative Hypothesis ($H_a$): Mean height $\neq$ 70 inches

• Statistical Method:

  • Statistic: Sample mean height of 30 people
  • Rejection Rule:
  • If $z<em>{\bar{x}} < z</em>0$, reject $H_0$.
  • If $z<em>{\bar{x}} > z</em>1$, reject $H_0$.
  • Step 1: Check if Samping Distribution is Normal.

• Decision-Making Process:

  • Collect a sample of height data and calculate the sample mean.
  • Calculate the $z$-score using the formula:
    $z = \frac{\bar{x} - 70}{\sigma / \sqrt{n}}$
  • Assess whether the calculated $z$-score falls within the critical values $z<em>0$ and $z</em>1$. If it falls outside this range, reject the null hypothesis.