HYPOTHESIS TESTING

Year 1 Edexcel Statistics: Hypothesis Testing

Definition: Hypothesis testing is a statistical method used to make decisions about the validity of a claim based on sample data. It involves two competing statements:

  1. Null Hypothesis (H0): The hypothesis that there is no effect or no difference, and it represents the status quo.

  2. Alternative Hypothesis (H1): The hypothesis that represents a new effect or difference that the researcher wants to prove.

Steps in Hypothesis Testing:

  1. State the Hypotheses:

    • Define the null and alternative hypotheses.

  2. Choose a Significance Level (α):

    • Common levels are 0.05, 0.01, or 0.10 which indicate the probability of rejecting the null hypothesis when it is true.

  3. Select the Appropriate Test:

    • Depending on the data, select a suitable test (e.g., t-test, z-test, chi-squared test).

  4. Calculate the Test Statistic:

    • Use sample data to compute the test statistic, which quantifies the difference between sample data and the null hypothesis.

  5. Determine the Critical Value or P-value:

    • A critical value defines a boundary for rejection of the null hypothesis. The p-value indicates the probability of observing the data if the null hypothesis is true.

  6. Make a Decision:

    • If the test statistic exceeds the critical value or if the p-value is less than the significance level, reject the null hypothesis. Otherwise, do not reject it.

  7. Conclusion:

    • Draw a conclusion based on the hypothesis test result in the context of the research question.

Example:

  • Suppose a researcher wants to test if a coin is biased towards heads.

    • H0: The coin is fair (p = 0.5)

    • H1: The coin is biased (p ≠ 0.5)

    • A sample of 100 flips shows 60 heads. The steps would involve calculating the test

An example question for Year 1 Hypothesis Testing in Edexcel A Level Maths could be:

"A manufacturer claims that their light bulbs have a lifespan of at least 1000 hours. To test this claim, a random sample of 30 light bulbs is tested, and the mean lifespan is found to be 950 hours with a standard deviation of 100 hours.

  1. State the null and alternative hypotheses.

  2. Choose a significance level, and explain why you chose this level.

  3. Select an appropriate test for this scenario and calculate the test statistic.

  4. Determine whether to reject or not reject the null hypothesis, and interpret the results in the context of the claim."

This question involves stating hypotheses, selecting the right statistical test, and calculating outcomes based on sample data. It tests understanding of the hypothesis testing process.

1. State the null and alternative hypotheses:

  • Null Hypothesis (H0): The mean lifespan of the light bulbs is at least 1000 hours (μ ≥ 1000).

  • Alternative Hypothesis (H1): The mean lifespan of the light bulbs is less than 1000 hours (μ < 1000).

2. Choose a significance level, and explain why you chose this level:A common significance level is α = 0.05. This means there is a 5% chance of rejecting the null hypothesis if it is actually true. This level is often used in hypothesis testing as it indicates a reasonable balance between Type I and Type II errors.

3. Select an appropriate test for this scenario and calculate the test statistic:Since the sample size is small (n = 30) and the population standard deviation is unknown, a t-test can be used. The formula for the t-test statistic is:

[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} ]Where:

  • (\bar{x} = 950) (sample mean)

  • (\mu_0 = 1000) (hypothesized mean)

  • (s = 100) (sample standard deviation)

  • (n = 30) (sample size)

Calculating the t-statistic:[ t = \frac{950 - 1000}{100 / \sqrt{30}} ][ t = \frac{-50}{18.26} \approx -2.74 ]

4. Determine whether to reject or not reject the null hypothesis, and interpret the results in the context of the claim:Using a t-distribution table, we can find the critical value for a one-tailed test with 29 degrees of freedom (n - 1) at a significance level of α = 0.05, which is approximately -1.699. Since our calculated t-statistic (-2.74) is less than -1.699, we reject the null hypothesis.

Interpretation: This result suggests that there is statistically significant evidence to conclude that the mean lifespan of the light bulbs is less than 1000 hours, supporting the claim that the manufacturer may not be meeting their lifespan assertions.