Question

At a particular university, students' grades in introductory statistic classes are generally unimodal and skewed to the left with a mean of u = 63 and a standard deviation of o = 16.2. (Round to 2 decimal places for all z-values and round all other answers to 4 decimal places, if needed.) (a) The distribution of students' grades is is approximately normal (b) If n = 33 students are selected at random, the distribution of the sample mean grade is approximately normal with a mean of and a standard deviation of (c) The probability that the sample mean grade for these 33 students is less than 66.0 is (d) If n = 33 students are selected at random, the distribution of the sample total grade is approximately normal with a mean of and a standard deviation of approximately normal not normal left-skewed (e) The probability that the total grade for these 33 students is less than 2178.0 is right-skewed

Answer 1

The distribution of students' grades is not approximately normal as it is skewed to the left. The distribution of the sample mean grade is approximately normal with the mean of the population mean and the standard deviation of the population standard deviation divided by the square root of the sample size.

Step-by-step explanation:

(a) The distribution of students' grades is not approximately normal because the given information states that the grades are unimodal and skewed to the left. The normal distribution is symmetric and bell-shaped, whereas a left-skewed distribution has a longer left tail.

(b) If n = 33 students are selected at random, the distribution of the sample mean grade is approximately normal with a mean of the population mean (u = 63) and a standard deviation of population standard deviation divided by the square root of the sample size (o/n = 16.2/√33).

(c) To find the probability that the sample mean grade for these 33 students is less than 66.0, we need to standardize the value of 66.0 using the sample mean and standard deviation, and then find the corresponding area under the standard normal curve using a Z-Table or a calculator.

(d) The distribution of the sample total grade is not approximately normal because the original distribution is left-skewed. In general, the total of values is affected by skewness and does not follow a normal distribution.

(e) Similarly, the probability that the total grade for these 33 students is less than 2178.0, assuming the original distribution is left-skewed, would depend on the specific shape of the distribution and cannot be easily calculated using the normal distribution.