For normal population mean 60, standard deviation 15, after treatment is given to the member of sample , the sample mean is 65, and sample size is 25.
is the sample mean sufficient to conclude that the treatment has significant effect? Give reason for decision?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. Is that less than the level of significance you are using?
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.