A researcher reports that "Students were randomly assignment to intervention or a control group; the two groups did not differ significantly prior to the start of the intervention. At the end of the study, those receiving the intervention showed statistically significant improvement (p < 0.05) in reading fluency, as compared to those in the control group." Which of the following is NOT a valid inference based on this statement.

A. We can infer that the samples come from two different populations at the end of the study.

B. We can assume that the two groups had identical means and standard deviations at the beginning of the study.

C.The probability that the two groups came from the same population was less than 5%.

D.There is about a 1 in 20 chance that the researcher is incorrectly inferring that the two groups, post-intervention, belong to different populations.

No, you can NOT infer that. I already answered this question I think.

The correct answer is B. We cannot assume that the two groups had identical means and standard deviations at the beginning of the study based on the given statement.

To understand why, let's break down the information provided in the statement:

- The researcher used random assignment to assign students to either an intervention group or a control group.
- Prior to the intervention, the two groups did not differ significantly. This means that there were no statistically significant differences between the groups in terms of reading fluency before the study began.
- At the end of the study, those in the intervention group showed statistically significant improvement in reading fluency compared to those in the control group. The significance level reported is p < 0.05, meaning that the probability of observing such a difference by chance alone is less than 5%.

Now, let's go through each option to determine which one is not a valid inference based on this statement:

A. We can infer that the samples come from two different populations at the end of the study.
This inference is valid because the statistically significant difference in reading fluency between the intervention and control groups suggests that they likely come from different populations in terms of reading abilities.

B. We can assume that the two groups had identical means and standard deviations at the beginning of the study.
This inference is not valid because although the statement mentions that the two groups did not differ significantly prior to the start of the intervention, it does not provide information about whether their means and standard deviations were identical.

C. The probability that the two groups came from the same population was less than 5%.
This inference is valid because the statement reports a statistically significant difference at a significance level of p < 0.05, indicating that the probability of observing such a difference by chance alone is less than 5%.

D. There is about a 1 in 20 chance that the researcher is incorrectly inferring that the two groups, post-intervention, belong to different populations.
This inference is valid because a significance level of p < 0.05 corresponds to a 1 in 20 chance of incorrectly inferring a difference between the groups when there is actually no true difference.

Therefore, the option that is NOT a valid inference based on the given statement is B. We cannot assume that the two groups had identical means and standard deviations at the beginning of the study.