8.3.2 Testing a Claim About a Mean - Sigma Known and When the Sampling Distribution Isn't Normal

This paragraph introduces the concept of testing a claim about a mean when the population standard deviation is known, especially in cases where the sample size is less than 30 and the population is not normally distributed. The speaker explains that despite these conditions, there are methods to test claims about the mean, using a different test statistic, 'z', from the standard normal distribution rather than the Student's t distribution. The formula for calculating the z-score is provided, which involves subtracting the population mean from the sample mean and dividing by the standard deviation of the sample means. The paragraph also outlines how to find p-values using z-scores and emphasizes the importance of checking the requirements for hypothesis testing before proceeding with the example of female attractiveness ratings in speed dating.

The speaker continues the discussion on hypothesis testing by demonstrating how to convert a sample mean into a z-score when the population standard deviation is known. The example involves the attractiveness ratings of men by female speed daters, with a sample size of 199, a sample mean of 3.91, and a sample standard deviation of 0.53. The null hypothesis is that the true population mean is 7.00, and the test is to determine if the sample mean is significantly lower than this value. The z-score calculation is detailed, resulting in a z-score of -21.9044, indicating that the sample mean is extremely low relative to the assumed population mean. The paragraph also explains the process of using p-values and critical values to make a decision about the null hypothesis, identifying this as a left-tailed test due to the 'less than' alternative hypothesis.

The speaker interprets the calculated z-score of -21.9044, emphasizing its extreme deviation from the mean, indicating a very low sample mean compared to the hypothesized population mean. The explanation of how to find the p-value using both a z-table and Excel is provided, with the p-value being extremely small (1.179 × 10^-106), suggesting that obtaining a sample mean as low as 3.91 is highly unlikely if the null hypothesis were true. The paragraph discusses the decision-making process for the null hypothesis, using both the p-value method and the critical value method, ultimately leading to the rejection of the null hypothesis due to the significant evidence against it.

The speaker acknowledges situations where the population is not normally distributed and the sample size is less than or equal to 30, and provides alternative options for testing claims about the mean. The paragraph briefly mentions bootstrap resampling and sign tests as methods that can be used in such cases, referring to additional resources for more detailed information. The speaker emphasizes that while these methods are not covered in the current course, they are important to be aware of and can be found in other sections of the textbook or in another book by the same author.

The speaker concludes the lesson on hypothesis testing for a mean with a known population standard deviation and provides a transition to future lessons. The conclusion emphasizes the rejection of the null hypothesis in the given example, supporting the claim that the mean rating is less than 7.00. The speaker also mentions that the next lessons will cover hypothesis tests about more than one proportion or more than one mean in chapter nine, indicating a continuation of the statistical topics.