Question

1.     T or F If a population is normally distributed, the sampling distribution of sample means will

also always be normally distributed

2.    If a population is not normally distributed when will the sampling distribution of sample means

be guaranteed to have a normal distribution?

3.    T or F For a given population and a given sample size there is only one sampling distribution of

sample mean that will be generated.   .

4.    The following 2 terms are NOT exactly the same. Define each including how they are similar and

how they are different

a.Sampling Error:  

b.    Standard Error:  

5.     T or F Another name for the standard deviation of the sampling distribution of sample means is

the standard error.  

6.      What are the 2 reasons that samples with larger sizes tend to have sample means closer to the

true population value?

7.     A population is normally distributed with a mean of 50 and a standard deviation of 20. For

samples of size 25, what is the probability of randomly sampling and find a sample mean of 54 or

more? Assume population is normally distributed. Show work for partial credit. Solve to a final

addition or subtraction step. Circle your final answer.

Answer 1

1. True: The sampling distribution of sample means will also always be normally distributed.

2. If the sample size is large enough so that CLT can be applied. In general, a sample size of 30 can be considered.

3. True: For a given population and a given sample size there can only be one sampling distribution of the sample mean that will be generated.

4. Sampling error: This is the error caused by observing a sample instead of the whole population. It is the difference between a sample statistic used to estimate a population parameter and the actual but unknown value of the parameter. This can also be seen as the error (in using the sample mean as an estimate of the true mean) that comes from the fact that we’ve chosen a random sample from the population, rather than chosen the entire population.

Standard error: This is essentially defined as the standard deviation of sample means around the population mean.

Difference between the two: Standard error is specifically the standard of sampling mean, but sampling error is for any statistic.

Similarity: Both are errors associated with a random sample from a population.

5. True: The standard deviation of the sampling distribution of the mean is called the standard error of the mean.

6. Law of Large Numbers and Central limit theorem.

7. Let X₁,X₂,....,X₂₅ is a random sample from Normal distribution with a mean of 50 and a standard deviation of 20. Then if denotes the sample mean. then   follows a normal distribution with mean and standard deviation 20/4 = 5.

Hence, the probability of sample mean of 54 or more is: