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Chapter 9: Problem 21

Cure for the Common Cold? An experiment was planned to compare the mean time (in days) required to recover from a common cold for persons given a daily dose of 4 milligrams (mg) of vitamin C versus those who were not given a vitamin supplement. Suppose that 35 adults were randomly selected for each treatment category and that the mean recovery times and standard deviations for the two groups were as follows: $$\begin{array}{lcc} & \begin{array}{l}\text { No Vitamin } \\\\\text { Supplement }\end{array} & \begin{array}{l}4 \mathrm{mg} \\\\\text { Vitamin }\end{array} \\\\\hline \text { Sample Size } & 35 & 35 \\\\\text { Sample Mean } & 6.9 & 5.8 \\\\\text { Sample Standard Deviation } & 2.9 & 1.2\end{array}$$

Short Answer

Expert verified
Explain your answer with the help of a two-sample t-test for independent samples and assuming the populations are normally distributed. Please provide the sample data for both groups including sample size, sample means, and sample standard deviations to perform the analysis.

Step by step solution

01

State the null and alternative hypotheses

The null hypothesis (H0) states that there is no difference in the mean recovery times between the two groups, and the alternative hypothesis (H1) states that there is a difference in the mean recovery times between the two groups. Mathematically: H0: 渭1 = 渭2 H1: 渭1 鈮� 渭2 where 渭1 is the population mean recovery time with no vitamin supplement, and 渭2 is the population mean recovery time with vitamin supplement.
02

Calculate the pooled variance and standard error

The pooled variance (s虏) and standard error (SE) are needed to calculate the t-score. The formulas are given as follows: s虏 = [(n1 - 1) * s1虏 + (n2 - 1) * s2虏] / (n1 + n2 - 2) SE = sqrt(s虏/n1 + s虏/n2) where n1 and n2 are the sample sizes, s1 and s2 are the sample standard deviations.
03

Calculate the t-score

The t-score can be calculated using the formula: t = (x虅1 - x虅2) / SE where x虅1 and x虅2 are the sample means for the two groups.
04

Calculate the degrees of freedom and find the critical t-value

The degrees of freedom can be calculated using the formula: df = n1 + n2 - 2 Then, look up the critical t-value in a t-distribution table with the given degrees of freedom and chosen significance level (e.g., 伪 = 0.05).
05

Compare the t-score with the critical t-value and determine the p-value

Compare the calculated t-score with the critical t-value, and determine if the null hypothesis can be rejected or not. If the t-score is greater than or equal to the critical t-value, we reject the null hypothesis and conclude that there is a significant difference in the mean recovery times for the two groups. We can also determine the p-value using a t-distribution table or statistical software, which represents the probability of observing this t-score or more extreme values when the null hypothesis is true. If the p-value is less than the chosen significance level (e.g., 0.05), we reject the null hypothesis.
06

Draw conclusions and interpret the results

Based on the comparison of the t-score with the critical t-value and the p-value, determine if there is a significant difference in the mean recovery times for the two groups. If the null hypothesis is rejected, we can conclude that taking a vitamin supplement does make a difference in the mean recovery time from a common cold. If we fail to reject the null hypothesis, we can conclude that there is insufficient evidence to claim that taking a vitamin supplement impacts the mean recovery time from a common cold.

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Most popular questions from this chapter

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