To determine if the sample mean of 65 is sufficient to conclude that the treatment has a significant effect, we need to conduct a hypothesis test.
Here's how you can do it:
Step 1: State the hypotheses.
- Null hypothesis (H0): The treatment has no significant effect. The population mean is still 60.
- Alternative hypothesis (Ha): The treatment has a significant effect. The population mean is different from 60.
Step 2: Set the significance level (α).
- The significance level is the probability of rejecting the null hypothesis when it is true. It is typically set at 0.05 (5%).
Step 3: Compute the test statistic.
- We'll use the z-test since we know the population standard deviation.
- The formula for the z-test is: (sample mean - population mean) / (population standard deviation / sqrt(sample size))
- Plugging in the values, we get: (65 - 60) / (15 / √25) = 5 / 3 = 1.67
Step 4: Find the p-value.
- The p-value is the probability of getting a test statistic value as extreme or more extreme than the one observed in the sample, assuming the null hypothesis is true.
- You can find the p-value using a z-table or a statistical software.
- In this case, the p-value associated with a z-score of 1.67 is 0.095 (approximately).
Step 5: Make a decision.
- Compare the p-value to the significance level (α) from Step 2.
- If the p-value is less than α, we reject the null hypothesis and conclude that the treatment has a significant effect. If the p-value is greater than α, we fail to reject the null hypothesis and do not conclude that the treatment has a significant effect.
- Since the p-value (0.095) is greater than α (0.05), we fail to reject the null hypothesis and conclude that the treatment does not have a significant effect.
In conclusion, based on the sample mean of 65 and the given information, we do not have enough evidence to conclude that the treatment has a significant effect on the population mean. However, it is important to note that this conclusion is specific to the given sample, and further investigation may be required for a more definitive conclusion.