Final answer:
The two values of x that contain the middle 65% of the normal curve area are most closely represented by the range that includes approximately 68% of the data, which is within one standard deviation from the mean. Therefore, the correct answer is (a) Between -1 and 1.
Step-by-step explanation:
The student asks about the two values of x that contain the middle 65% of the normal curve area. To find these two values, we refer to the Empirical Rule (or 68-95-99.7 rule) for normal distributions, which states that approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.Since the student is asking for the middle 65%, this does not match exactly to any of the well-known percentages from the Empirical Rule, but it is closer to 68% than to 95% or 99.7%. Therefore, the range that includes approximately 68% of the data within one standard deviation is the closest fit. We know from the rule that one standard deviation from the mean in either direction on the normal curve accounts for about 68% of the data. Hence, the values of x would be between -1 and +1 standard deviations. This can be represented as answer choice (a) Between -1 and 1.
The two values of x that contain the middle 65% of the normal curve area are:a) Between -1 and 1To determine this, we need to find the z-scores that correspond to the lower and upper boundaries of the 65% area. Since the normal distribution is symmetric, we can use the standard normal distribution table to find these z-scores. The z-score for the lower boundary is found by subtracting the standard deviation multiplied by the z-score (which corresponds to the given area) from the mean. The z-score for the upper boundary is found by adding the standard deviation multiplied by the z-score from the mean. In this case, the z-scores for the lower and upper boundaries of the 65% area are approximately -0.3853 and 0.3853, respectively. Converting these z-scores back to the original values of x, we get: -1 and 1.