To find out whether Medicine X is more effective at managing insomnia than Medicine Y, 11 patien

step 1 For this problem, we are told that to find out whether medicine X is more effective at managing

To find out whether Medicine X is more effective at managing...

To find out whether Medicine $\mathrm{X}$ is more effective at managing insomnia than Medicine $Y, 11$ patients are selected. 6 receive Medicine $X$, and 5 receive Medicine Y. The time taken to fall asleep, in minutes, is as follows: Medicine X: 22, 24, 18, 28, 16, 20 Medicine $Y: 18,20,14,20,14$ At the 0.05 level of significance, can medicine $Y$ be said to be more effective? Assume the two populations are normally distributed with equal variances.

To find out whether Medicine $\mathrm{X}$ is more effective at managing insomnia than Medicine $Y, 11$ patients are selected. 6 receive Medicine $X$, and 5 receive Medicine Y. The time taken to fall asleep, in minutes, is as follows:
Medicine X: 22, 24, 18, 28, 16, 20
Medicine $Y: 18,20,14,20,14$

At the 0.05 level of significance, can medicine $Y$ be said to be more effective? Assume the two populations are normally distributed with equal variances.

Key Concepts

Significance Level and p-Value

The significance level (often set at 0.05) defines the threshold for deciding whether an observed result is statistically significant. The p-value, which is compared against this threshold, indicates the probability of obtaining the observed result (or one more extreme) if the null hypothesis were true. A p-value lower than the significance level typically leads to rejection of the null hypothesis.

Assumptions of Normality and Equal Variance

Many inferential tests, including the two-sample t-test, assume that the data from each group are normally distributed and that the two populations have equal variances. These assumptions are crucial for the validity of the test results and the proper application of the t-distribution.

Null and Alternative Hypotheses

The null hypothesis represents the status quo or a statement of no effect, while the alternative hypothesis represents the claim being tested, such as the existence of a significant difference between two groups. The decision to reject or fail to reject the null hypothesis is made based on the evidence provided by the sample data.

Hypothesis Testing

A method of statistical inference that involves formulating a null hypothesis and an alternative hypothesis, then using sample data to decide whether there is sufficient evidence to reject the null hypothesis. It helps in assessing if an observed effect is statistically significant or could have occurred by chance.

Two-Sample t-Test

A statistical test used to compare the means of two independent samples to determine if there is a statistically significant difference between them. It is particularly useful when the sample sizes are small and the population variances are assumed to be equal.

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