Hypothesis Testing Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

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A school claims its students average 75 on standardized tests. A sample of n=36n=36 gives xห‰=78\bar{x}=78 with ฯƒ=12\sigma=12. Test H0:ฮผ=75H_0: \mu=75 vs Ha:ฮผ>75H_a: \mu>75 at ฮฑ=0.05\alpha=0.05.

Solution

  1. 1
    Calculate test statistic: z=xห‰โˆ’ฮผ0ฯƒ/n=78โˆ’7512/36=32=1.5z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} = \frac{78 - 75}{12/\sqrt{36}} = \frac{3}{2} = 1.5
  2. 2
    Find p-value (one-tailed): P(Z>1.5)=1โˆ’0.9332=0.0668P(Z > 1.5) = 1 - 0.9332 = 0.0668
  3. 3
    Compare to ฮฑ=0.05\alpha = 0.05: p=0.0668>0.05p = 0.0668 > 0.05
  4. 4
    Decision: Fail to reject H0H_0. Conclusion: insufficient evidence that the true mean exceeds 75.

Answer

z=1.5z=1.5, p=0.067>0.05p=0.067 > 0.05. Fail to reject H0H_0. Evidence is inconclusive.
Hypothesis testing follows a structured procedure: state hypotheses, calculate test statistic, find p-value, compare to ฮฑ, state conclusion. Failing to reject Hโ‚€ does not prove Hโ‚€ is true โ€” it means the data is insufficient to reject it.

About Hypothesis Testing

A systematic method to decide whether sample data provides enough evidence to reject a claim (null hypothesis) about a population parameter.

Learn more about Hypothesis Testing โ†’

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