5.3.2 Normal Hypothesis Testing

Normal hypothesis testing is a statistical method used to determine whether a population parameter (such as the mean) differs from a hypothesised value. This method uses the Normal distribution as a basis for making inferences about the population. It's particularly useful when dealing with large sample sizes where the Central Limit Theorem applies, making the sampling distribution of the mean approximately normal.

Key Concepts in Hypothesis Testing

Null Hypothesis ( $H_0$ ): The assumption that there is no effect or no difference. It's the hypothesis that we aim to test against.
Alternative Hypothesis ( $H_1$ ): The assumption that there is an effect or a difference. It is what you conclude if you reject the null hypothesis.
Significance Level ( $\alpha$ ): The probability of rejecting the null hypothesis when it is actually true. Common values are 0.05 (5%) or 0.01 (1%).
Test Statistic: A standardised value that is calculated from sample data during a hypothesis test.
P-Value: The probability of obtaining a result at least as extreme as the observed result, assuming that the null hypothesis is true.
Critical Value: The threshold value that the test statistic must exceed to reject the null hypothesis.

infoNote

Example: Testing the Mean Suppose a company claims that their light bulbs last an average of 1,000 hours. A consumer group believes that the true mean lifetime is less than 1,000 hours and wants to test this claim. They take a random sample of 50 bulbs and find an average lifetime of 980 hours with a standard deviation of 40 hours. They wish to test this at the 5% significance level.

Step 1: State the Hypotheses

Null Hypothesis ( $H_0$ ): $\mu = 1000$ (The mean lifetime is 1,000 hours.)
Alternative Hypothesis ( $H_1$ ): $\mu < 1000$ (The mean lifetime is less than 1,000 hours.)

Step 2: Choose the Significance Level

Significance level ( $\alpha$ ): 0.05

Step 3: Calculate the Test Statistic Since the sample size is large, the sampling distribution of the mean can be approximated by a Normal distribution. The test statistic $(z)$ is calculated as:

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

Where:

$\bar{x} = 980$ (sample mean)
$\mu_0 = 1000$ (hypothesised population mean)
$\sigma = 40$ (sample standard deviation)
$n = 50$ (sample size) Substitute the values:

z = \frac{980 - 1000}{\frac{40}{\sqrt{50}}} = \frac{-20}{\frac{40}{7.071}} \approx -3.54

Step 4: Find the Critical Value or P-Value For a significance level of 0.05 in a one-tailed test, the critical $z-value$ is approximately -1.645.

Alternatively, you can find the $p-value$ associated with the test statistic:

Using the $z-table$ , $P(Z < -3.54)$ is very small, approximately 0.0002.

Step 5: Make a Decision

Since the test statistic $z = -3.54$ is less than the critical value of -1.645, we reject the null hypothesis.
Alternatively, because the $p-value$ (0.0002) is less than the significance level (0.05), we reject the null hypothesis. Conclusion: There is strong evidence at the 5% significance level to suggest that the mean lifetime of the light bulbs is less than 1,000 hours.

Conclusion

Normal hypothesis testing is a vital tool in statistics for making inferences about population parameters based on sample data. By following the steps of stating hypotheses, calculating the test statistic, and comparing it to a critical value or $p-value$ , you can make informed decisions about the validity of claims. Practising these steps with different examples will help you become confident in performing hypothesis tests.

Normal Hypothesis Testing Simplified Revision Notes for A-Level AQA Maths Statistics

5.3.2 Normal Hypothesis Testing

Key Concepts in Hypothesis Testing

Conclusion

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