Final answer:
Without an explicit p-value, we infer that if the p-value were less than 0.05, we would reject the null hypothesis and conclude that the new painkiller brings faster relief. However, a definitive decision requires calculating the actual p-value, which is not provided in this scenario.
Step-by-step explanation:
To determine whether the new painkiller brings faster relief compared to the standard painkiller with an average relief time of 3.5 minutes, we conduct a hypothesis test. We are given that a sample of 40 patients had a mean relief time of 3.0 minutes with a standard deviation of 1.1 minutes. We will test the null hypothesis, H0: μ ≥ 3.5, against the alternative hypothesis, Ha: μ < 3.5, at a significance level (α) of 0.05.
The test statistic for this hypothesis test follows a t-distribution due to the small sample size and unknown population standard deviation. However, since the exact p-value is not provided, and we need to make a decision based on the information, we can only assume the standardized form of hypothesis testing guidelines. If the p-value is less than the significance level, we reject the null hypothesis. If the p-value is greater, we fail to reject it.
Given the mean of the sample is less than the population mean under the null hypothesis, this suggests we may have to reject the null hypothesis if the p-value is indeed smaller than 0.05. However, without the actual calculation of the p-value, the definitive decision cannot be conclusively reached from the information provided. Normally, one would calculate the p-value using the sample statistics and the t-distribution to reach a decision.
Giving a professional assessment based on typical statistical analysis, if the p-value were less than 0.05, the correct decision would most likely be option a, Reject the null hypothesis; the new painkiller brings faster relief.