Question

In a study of heart surgery, one issue was the effect of drugs called beta-blockers on the pulse rate of patients during surgery. The available subjects were divided at random into two groups of 30 patients each. One group received a beta-blocker; the other group received a placebo. The pulse rate of each patient at a critical point during the operation was recorded. The treatment group had a mean pulse rate of 65.2 and standard deviation 7.8. For the control group, the mean pulse rate was 70.3 and the standard deviation was 8.3. What is the probability that the difference in the samples (treatment group – placebo group) is greater than 0?

Answer 1

To determine the probability that the treatment group's mean pulse rate is greater than that of the placebo group, we would use a two-sample t-test. This requires the calculation of the t-statistic and the subsequent p-value, which is not computable without additional information, although the methodology for the hypothesis testing is described.

Step-by-step explanation:

The student has presented a scenario involving an experiment to determine the effect of beta-blockers on patient pulse rate during heart surgery. The main question posed is: What is the probability that the difference in mean pulse rates between the treatment group and the placebo group (treatment group – placebo group) is greater than 0? This question concerns the concepts of statistics and hypothesis testing. To answer this question, one would typically employ a two-sample t-test to compare the means of two independent groups. Given the provided means, standard deviations, and sample sizes for both groups, it is possible to compute the t-statistic and subsequently find the p-value associated with this statistic. The p-value would provide the probability that the difference in sample means is greater than 0 due to random chance, under the assumption that the null hypothesis (no difference between the groups) is true.

Since pulse rates are normally distributed, we can indeed apply the two-sample t-test. We are comparing the mean pulse rate of one group that received a beta-blocker with another that received a placebo. We calculate the t-statistic using the formula:

1. Calculate the difference in sample means (Δμ = μ1 - μ2).
2. Calculate the standard error of the difference in means (SEΔμ).
3. Compute the t-statistic (t = Δμ / SEΔμ).
4. Compare the t-statistic with the critical value from the t-distribution table or use statistical software to find the p-value.

If the p-value is less than the chosen significance level (commonly 0.05), then we reject the null hypothesis, indicating a statistically significant difference between the groups. However, with the information provided, the actual calculations and resulting probability cannot be computed without further details, such as the actual mean differences and the standard error of the difference. Still, the methodology described here outlines the steps required to determine the likelihood that the observed difference in sample means is due to the effect of the beta-blockers given the provided data.