Q. 3.219 - Elementary Statistics

Question

Consider the following three data sets.

Part (a): Assuming that each of these data sets is sample data, compute the standard deviations. (Round the answers to two decimals places)

Part (b): Assuming that each of these data sets is population data, compute the standard deviations. (Round the answers to two decimals places)

Part (c): Using your results from parts (a) and (b), make an educated guess about the answer to the following question: If both s and $\sigma$ are computed for the same data set, will they tend to be closer together if the data set is large or if it is small?

Step-by-Step Solution

Answer

Part (a): The sample standard deviation for data set 1 is 2.16.

The sample standard deviation for data set 2 is 2.04.

The sample standard deviation for data set 3 is 2.01.

Part (b): The population standard deviation for data set 1 is 1.87.

The population standard deviation for data set 2 is 1.88.

The population standard deviation for data set 3 is 1.91.

Part (c): If both s and $\sigma$are computed for the same data set, sample standard deviation is inversely proportion to sample size where as population standard deviation is direct proportion to sample size.

Part (a) Step 1. Given information.

Consider the given question,

Part (a) Step 2. Write the descriptive statistics for the given data.

The descriptive statistics for the given data set is given below,

Part (a) Step 3. Compute the standard deviations for sample data.

Using excel,

The sample standard deviation for data set 1 is 2.16.

Therefore, $s_1=2.16$

The sample standard deviation for data set 2 is 2.04.

Therefore, $s_2=2.04$

The sample standard deviation for data set 3 is 2.01.

Therefore, $s_3=2.01$

Part (b) Step 1. Compute the standard deviations for population data.

Using excel,

The population standard deviation for data set 1 is 1.87.

Therefore, ${\sigma }_1=1.87$

The population standard deviation for data set 2 is 1.88.

Therefore, ${\sigma }_2=1.88$

The population standard deviation for data set 3 is 1.91.

Therefore, ${\sigma }_3=1.91$

Part (c) Step 1. Compare the results of parts (a) and (b).

Consider the parts (a) and (b),

We observe the sample deviation decreases when the data points increases and population standard deviation increases when data points increases.

Sample standard deviation is inversely proportion to sample size where as population standard deviation is direct proportion to sample size.