In a study of performance ratings of ex-smokers, a random...

In a study of performance ratings of ex-smokers, a random sample of 34 ex-smokers had a mean rating of 2.21 and a sample standard deviation of 2.21. For an independent random sample of 86 long-term ex-smokers, the mean rating was 1.47 and the sample standard deviation was 1.69 . Find the lowest level of significance at which the null hypothesis of equality of the two population means can be rejected against a two-sided alternative.

Key Concepts

p-value

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value assuming the null hypothesis is true. In hypothesis testing, if the p-value is less than or equal to the significance level, the null hypothesis is rejected, indicating that the results are statistically significant.

Significance Level

The significance level, denoted by alpha (?), is the threshold probability used in hypothesis testing to decide whether to reject the null hypothesis. It represents the maximum probability of committing a Type I error, which occurs when the null hypothesis is incorrectly rejected when it is actually true.

Test Statistic

The test statistic is a numerical value calculated from the sample data, which, in a two-sample t-test, measures the size of the difference relative to the variation in the sample data. It quantifies how many standard errors the observed difference is away from the null hypothesis expectation.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in the data that can vary when estimating statistical parameters. In a two-sample t-test, they are used to determine the appropriate t-distribution and are typically calculated from the sample sizes, affecting the shape of the distribution used for significance testing.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis. Here, it suggests that the two population means are not equal, leading to a two-sided test where differences in either direction are considered significant.

Null Hypothesis

The null hypothesis is a statement that there is no effect or no difference between groups—in this context, it asserts that the two population means are equal. It serves as the baseline assumption that any observed differences in sample data are due to random sampling variability.

Two-Sample t-Test

A two-sample t-test is a statistical method used to compare the means of two independent groups when the population standard deviations are unknown. It takes into account the sample means, sample standard deviations, and sample sizes, using a t-distribution to determine the likelihood that any observed difference occurred by chance.

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