A researcher administered a treatment to a sample of participants selected from a population

of 80. If the researcher obtains a sample mean of M=88. Which combination of factors is
most likely to result in rejecting the null hypothesis?
a, alpha level=0.01 and standard deviation=5
b, alpha level=0.05 and standard deviation=5
c, alpha level=0.01 and standard deviation=10
d, alpha level=0.05 and standard deviation=10

B, the largest alpha level and the smallest SD.

To determine which combination of factors is most likely to result in rejecting the null hypothesis, we need to consider the significance level (alpha level) and the standard deviation.

The null hypothesis is typically rejected when the sample mean falls outside of the range defined by the critical values. The critical values are based on the significance level and the standard deviation.

Let's analyze the given options:

a) alpha level = 0.01 and standard deviation = 5
b) alpha level = 0.05 and standard deviation = 5
c) alpha level = 0.01 and standard deviation = 10
d) alpha level = 0.05 and standard deviation = 10

Considering that the sample mean is 88, we can calculate the critical values for each option and determine which combination falls outside of that range.

a) With an alpha level of 0.01 and standard deviation of 5, the critical values would be 88.71 (88 + 1.645*(5/sqrt(80))) and 87.29 (88 - 1.645*(5/sqrt(80))). Since the sample mean of 88 falls within this range, the null hypothesis would not be rejected.

b) With an alpha level of 0.05 and standard deviation of 5, the critical values would be 88.34 (88 + 1.96*(5/sqrt(80))) and 87.66 (88 - 1.96*(5/sqrt(80))). Again, since the sample mean of 88 falls within this range, the null hypothesis would not be rejected.

c) With an alpha level of 0.01 and standard deviation of 10, the critical values would be 88.43 (88 + 2.576*(10/sqrt(80))) and 87.57 (88 - 2.576*(10/sqrt(80))). The sample mean of 88 still falls within this range, so the null hypothesis would not be rejected.

d) With an alpha level of 0.05 and standard deviation of 10, the critical values would be 88.17 (88 + 1.96*(10/sqrt(80))) and 87.83 (88 - 1.96*(10/sqrt(80))). Once again, the sample mean of 88 falls within this range, so the null hypothesis would not be rejected.

Therefore, none of the given combinations of factors are likely to result in rejecting the null hypothesis in this scenario.

To determine which combination of factors is most likely to result in rejecting the null hypothesis, we need to consider the sample mean, the alpha level (significance level), and the standard deviation.

The null hypothesis (H0) assumes that there is no significant difference between the treatment and the population mean. The alternative hypothesis (Ha) assumes that there is a significant difference.

To test the null hypothesis, we typically use a hypothesis test, such as a t-test. The t-test compares the sample mean to the population mean and takes into account the sample standard deviation.

The alpha level (significance level) is the probability of falsely rejecting the null hypothesis. It represents the acceptable level of risk for rejecting a true null hypothesis. Common alpha levels are 0.05 and 0.01, where 0.05 allows for a higher chance of rejecting the null hypothesis compared to 0.01.

Higher standard deviation indicates more variability in the data, which can make it more difficult to detect significant differences or effects.

Now, let's analyze the given combinations of factors:

a) Alpha level = 0.01 and standard deviation = 5
b) Alpha level = 0.05 and standard deviation = 5
c) Alpha level = 0.01 and standard deviation = 10
d) Alpha level = 0.05 and standard deviation = 10

In order to reject the null hypothesis, we need to find evidence that the sample mean is significantly different from the population mean. This could be more likely if the sample mean is very different from the population mean, or if the alpha level is set at a lower value allowing for less chance of a false positive.

Comparing the combinations:

a) With an alpha level of 0.01, there is a lower chance of falsely rejecting the null hypothesis compared to 0.05. However, with a standard deviation of 5, there is not much variability in the data. Therefore, this combination is less likely to result in rejecting the null hypothesis.

b) With an alpha level of 0.05, there is a higher chance of falsely rejecting the null hypothesis compared to 0.01. The standard deviation of 5 indicates less variability in the data. This combination could be more likely to result in rejecting the null hypothesis.

c) With an alpha level of 0.01 (lower chance of false positive) and a standard deviation of 10 (more variability in the data), this combination might provide a better chance of finding significant differences and rejecting the null hypothesis.

d) With an alpha level of 0.05 and a standard deviation of 10, this combination may also have a higher chance of resulting in rejecting the null hypothesis, but the higher alpha level increases the probability of making a Type I error (false rejection of the null hypothesis).

Based on the analysis, options c (alpha level = 0.01 and standard deviation = 10) and d (alpha level = 0.05 and standard deviation = 10) are more likely to result in rejecting the null hypothesis. However, option c provides a lower chance of a false positive due to the lower alpha level and may be preferred.