Question

Consider measuring the time until death after diagnosis of a particular disease, and suppose that we have complete follow up, so that all death times are observed. We will assume the following to be known values for the population. We know that the mean time until death after diagnosis is 6.5 years. We also know that the standard deviation of time until death is 4.1 years. Further, we know that the distribution of time until death is very strongly skewed to the right (positively skewed) for this population. Use this to answer the following.

Part (a) ( WebWorkiR ) Suppose that we take a simple random sample of 15 people from this population. What would we 'expect' our sample mean to be for our sample of 15 individuals? That is, what is the expected value of the sample mean?

A. It depends on the sample size
B. Less than 6.5
C. Exactly equal to 6.5
D. Greater than 6.5


Part (b) ( WebWorkiR ) Suppose that we take a simple random sample of 15 people from this population. What would we 'expect' our sample standard deviation to be for our sample of 15 individuals? That is, what is the expected value of the sample standard deviation?

A. Less than 4.1
B. Greater than 4.1
C. It depends on the sample size
D. Exactly equal to 4.1


Part (c) ( WebWorkiR ) If we made a histogram of these 10 observations, what shape would we 'expect' this histogram to have?

A. It depends on the sample size
B. Skewed to the right
C. Skewed to the left
D. Normally distributed
E. Symmetric


Part (d) ( WebWorkiR ) Suppose that we take a simple random sample of 150 people from this population. What would we 'expect' our sample mean to be for our sample of 150 individuals? That is, what is the expected value of the sample mean?

A. Exactly equal to 6.5
B. Greater than 6.5
C. It depends on the sample size
D. Less than 6.5


Part (e) ( WebWorkiR ) Suppose that we take a simple random sample of 150 people from this population. What would we 'expect' our sample standard deviation to be for our sample of 150 individuals? That is, what is the expected value of the sample standard deviation?

A. Less than 4.1
B. Exactly equal to 4.1
C. It depends on the sample size
D. Greater than 4.1


Part (f) ( WebWorkiR ) If we made a histogram of these 150 observations, what shape would we 'expect' this histogram to have?

A. It depends on the sample size
B. Skewed to the right
C. Skewed to the left
D. Symmetric
E. Normally distributed


Part (g) ( On-Paper ) Again, consider taking a sample of 150 observations from this population. In a few sentences (you are encouraged to use pictures or notation if this helps), explain what is mean by the term "The Sampling Distribution of The Sample Mean", in the context of this example. In your explanation, imagine that you are explaining to a friend with limited understanding of statistics, and avoid the use of technical terms as much as possible. Also, please state any assumptions that are relevant to your description of the sampling distribution

Answer 1

Part(a) We know sample mean is an unbiased estimator of population mean. Therefore, the sample mean will be excatly equal to the population mean.

ANS: C. Excatly equal to 6.5

Part(b) We know sample standard deviation is an unbiased estimator of population standard deviation. Therefore, the sample standard deviation will be excatly equal to the population standard deviation.

ANS: D. Excatly equal to 4.1

Part(c) Since the original distribution is very strongly skewed to the right; and the number of observations is 10 that is less than 15 .

Therefore, the histogram is also expected to be skewed to the right.

ANS: B. Skewed to the right

Part(d) We know sample mean is an unbiased estimator of population mean. Therefore, the sample mean will be excatly equal to the population mean.

ANS: A. Excatly equal to 6.5

Part(e) We know sample standard deviation is an unbiased estimator of population standard deviation. Therefore, the sample standard deviation will be excatly equal to the population standard deviation.

ANS: B. Excatly equal to 4.1

Part(f) Though the original distribution is very strongly skewed to the right but,we have observations is 150 that is more than 40. Therefore, the histogram is expected to be normally distributed.

ANS: E. Normally distributed

Part(g) A sampling distribution of a sample mean is actually a probability distribution of the sample mean obtained from the samples drawn from the population or, in other words the sampling distribution of the mean is the distribution of the sample mean considered as a random variable( a variable whose values depend on outcomes of a random phenomenon), when derived from a random sample of size 'n' (in this case n =150). The sampling distribution depends on the underlying distribution of the population and the sample size. Since, the sample size is 150 which is large enough we can say the sampling distribution of sample mean will be normal (from central limit theorem).

Assumptions: 1. The sampled values must be independent of each other.

2. The data values must be sampled randomly.