For a random sample of 125 British entrepreneurs, the mean...

For a random sample of 125 British entrepreneurs, the mean number of job changes was 1.91 and the sample standard deviation was 1.32 . For an independent random sample of 86 British corporate managers, the mean number of job changes was 0.21 and the sample standard deviation was 0.53 . Test the null hypothesis that the population means are equal against the alternative that the mean number of job changes is higher for British entrepreneurs than for British corporate managers.

Key Concepts

Assumptions

Key assumptions for the two-sample t-test include the independence of observations between groups, approximate normality of the data in each group, and, if pooling variances, the assumption that the population variances are equal. If these assumptions are not met, alternative statistical methods or adjustments may be required.

Test Statistic Calculation

The test statistic for a two-sample t-test is calculated by taking the difference between the sample means and dividing it by the standard error of the difference. This standard error is computed from the sample standard deviations and sample sizes, and the resulting statistic follows a t-distribution under the null hypothesis.

p-value and Statistical Significance

The p-value quantifies the probability of observing the test results, or results more extreme, assuming the null hypothesis is true. A low p-value indicates that the observed difference is unlikely to occur by chance alone, leading to the rejection of the null hypothesis and suggesting a statistically significant difference.

Two-Sample t-Test

The two-sample t-test is a statistical method for comparing the means of two independent groups. It determines whether the difference between the sample means is statistically significant, taking into account the variation within each sample by using their standard deviations and sample sizes.

Hypothesis Testing

Hypothesis testing is a statistical framework used to assess whether observed data deviates significantly from what is expected under a stated null condition. It involves formulating a null hypothesis (usually indicating no effect or no difference) and an alternative hypothesis, then using sample data to determine if there is sufficient evidence to reject the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H0) typically posits that there is no difference between the population means, while the alternative hypothesis (H1) suggests a directional or non-directional difference exists. Correctly setting up these hypotheses is fundamental as they define the statistical test and interpretation of results.

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