Spoken Words Couples were recruited for a study of how many...

Spoken Words Couples were recruited for a study of how many words people speak in a day. A random sample of 56 males resulted in a mean of 16,576 words and a standard deviation of 7871 words. Use a $0.01$ significance level to test the claim that males have a standard deviation that is greater than the standard deviation of 7460 words for females (based on Data Set $24^{\text {"'Word Counts" }}$.

Key Concepts

Significance Level

The significance level, often denoted as alpha, is the threshold used to determine when to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. In hypothesis testing, if the p-value is less than the significance level, the null hypothesis is rejected in favor of the alternative.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that are free to vary when calculating statistical estimates. In the context of the chi-square test for variance, the degrees of freedom are typically calculated as the sample size minus one, and they are used to determine the appropriate chi-square distribution for finding critical values for the test.

p-Value

The p-value is the probability of obtaining test results at least as extreme as those observed during the test, assuming that the null hypothesis is correct. A low p-value indicates that the observed data are unlikely under the null hypothesis, which can lead to its rejection when the p-value falls below the predetermined significance level.

Hypothesis Testing

Hypothesis testing is a statistical process used to determine whether there is enough evidence in a sample to support a claim about a population parameter. It involves formulating a null hypothesis that represents a baseline or status quo and an alternative hypothesis that represents a new claim to be tested, then using sample data to assess the validity of the null hypothesis at a predetermined significance level.

Chi-Square Test for Variance

The chi-square test for variance is used to evaluate claims about the variability in a normally distributed population. The test statistic is calculated by multiplying the sample variance by the degrees of freedom (which is the sample size minus one) and dividing by the hypothesized variance. This statistic follows a chi-square distribution, and its value is compared to critical values from the chi-square distribution to decide whether to reject the null hypothesis.

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