Assignment 1: Answering Questions about Probability Using Z Scores
One way to use the second proportion of each z score as a p value is when comparing databases to see if a data point is more likely in one than the other.
To see how this works, generate a second normal distribution and compare the p values of the same data points as above (33, 35, 43, 22, 17) when µ = 25 and σ = 5 in Figure 2 to the p values they take on when µ = 25 and σ = 8 (same mu, different sigma). Generate the new set of z scores for the data points and compare the corresponding probabilities. Use a blank bell curve to plot µ, calculate the new values for the data at each whole number z score and the location of each data point (33, 35, 43, 22, 17) so you can see how location varies depending on sigma (σ).
For example, when µ = 25 and σ = 5 and a data point of 30, z = 1.00 and p = .160. However, when µ = 25 and σ = 8, the same data point of 30 has z = 0.63 and p = .260. Because z = (30-25)/8 = 0.63, the associated proportions/probabilities from z-score table are .2357 and .2643, respectively. Rounding off the second proportion to three decimal places, p = .260.
Now generate a third distribution, but this time reduce the scatter of data: µ = 25 and σ = 2. Using the same data points from above (33, 35, 43, 22, 17), generate z scores for and compare the corresponding probabilities.
For example, when µ = 25 and σ = 2, the data point of 30 has z = 2.50 and p = .010. Because z = (30-25)/2 = 2.50, the associated proportions/probabilities from z-score table are .4938 and .0062, respectively. Rounding off the second proportion to three decimal places, p = .010.
Use the data in the table below for the following 4 problems. For each, use the stated µ and σ to calculate the z score, get the p value (3 decimal places), and write out a formal APA statement of conclusion for each child. When you are finished with that, then answer the following question for each pair of parameters: Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis? What happens to the “significance” of each child’s data as the data are progressively more dispersed?
NOTE: Problems #1 and #2 are PRACTICE PROBLEMS with the answers available via the link beside them.
Problems #3 and #4 are GRADED PROBLEMS
Please submit all 4 problems using the Module 3 Assignment 1 Template found in the Doc Sharing area.
Problem 1. µ = 100 seconds and σ = 10 (practice problem – link to answer)
Problem 2. µ = 100 seconds and σ = 20 (practice problem – link to answer)
Problem 3. µ = 100 seconds and σ = 30 (graded problem)
Problem 4. µ = 100 seconds and σ = 40 (graded problem)
Given the research scenario, data points, set of population parameters and alpha set at p = .05, is the student able to generate the correct:
•Pair of hypotheses for each data point
•A z statistic and p value for each data point
•Decision about the null hypothesis for each data point
•APA-formatted statement of results for each data point
Please report z scores to two decimals and p values to three decimals. If the p value is less than .001, report it as p < .001.
Module 3 Assignment 1 Template
Fill in the answers for each table and answer the concluding FOUR questions below. Please round up your z scores to the hundredths spot, two decimals to the right and p values to the thousandths spot three decimals to the right as needed. If the p value is less than .001, please report p < .001. Please note when a value appears in SPSS as .000, it is truly p<.001. If you click on the .000 in the SPSS output you can see the value.
Make sure to write out a formal APA statement of conclusion for each child as the instructions state.
________________________________________
PROBLEM #1
Problem 1: µ = 100 ; σ =10
Child Concentration Z P Hypothesis Error
Child 1 75 z= p= Decision about H0: Possible Error:
APA Statement:
Child 2 81 z= p= Decision about H0: Possible Error:
APA Statement:
Child 3 89 z= p= Decision about H0: Possible Error:
APA Statement:
Child 4 99 z= p= Decision about H0: Possible Error:
APA Statement:
Child 5 115 z= p= Decision about H0: Possible Error:
APA Statement:
Child 6 127 z= p= Decision about H0: Possible Error:
APA Statement:
Child 7 138 z= p= Decision about H0: Possible Error:
APA Statement:
Child 8 139 z= p= Decision about H0: Possible Error:
APA Statement:
Child 9 142 z= p= Decision about H0: Possible Error:
APA Statement:
Child 10 148 z= p= Decision about H0: Possible Error:
APA Statement:
Report z scores to two decimals and p values to three decimals. If the p value is less than .001, report it as p < .001.
Problem 1 µ = 100
σ =10
Child Concentration z score p value
1 75
2 81
3 89
4 99
5 115
6 127
7 138
8 139
9 142
10 148
Add your 10 APA-style conclusion statements for Problem 1 here.
Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?
________________________________________
PROBLEM #2
Problem 2: µ = 100 ; σ =20
Child Concentration Z P Hypothesis Error
Child 1 75 z= p= Decision about H0: Possible Error:
APA Statement:
Child 2 81 z= p= Decision about H0: Possible Error:
APA Statement:
Child 3 89 z= p= Decision about H0: Possible Error:
APA Statement:
Child 4 99 z= p= Decision about H0: Possible Error:
APA Statement:
Child 5 115 z= p= Decision about H0: Possible Error:
APA Statement:
Child 6 127 z= p= Decision about H0: Possible Error:
APA Statement:
Child 7 138 z= p= Decision about H0: Possible Error:
APA Statement:
Child 8 139 z= p= Decision about H0: Possible Error:
APA Statement:
Child 9 142 z= p= Decision about H0: Possible Error:
APA Statement:
Child 10 148 z= p= Decision about H0: Possible Error:
APA Statement:
Report z scores to two decimals and p values to three decimals. If the p value is less than .001, report it as p < .001.
Problem 2 µ = 100
σ =20
Child Concentration z score p value
1 75
2 81
3 89
4 99
5 115
6 127
7 138
8 139
9 142
10 148
Add your 10 APA-style conclusion statements for Problem 2 here.
Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?
________________________________________
PROBLEM #3
Problem 3: µ = 100 ; σ =30
Child Concentration Z P Hypothesis Error
Child 1 75 z= p= Decision about H0: Possible Error:
APA Statement:
Child 2 81 z= p= Decision about H0: Possible Error:
APA Statement:
Child 3 89 z= p= Decision about H0: Possible Error:
APA Statement:
Child 4 99 z= p= Decision about H0: Possible Error:
APA Statement:
Child 5 115 z= p= Decision about H0: Possible Error:
APA Statement:
Child 6 127 z= p= Decision about H0: Possible Error:
APA Statement:
Child 7 138 z= p= Decision about H0: Possible Error:
APA Statement:
Child 8 139 z= p= Decision about H0: Possible Error:
APA Statement:
Child 9 142 z= p= Decision about H0: Possible Error:
APA Statement:
Child 10 148 z= p= Decision about H0: Possible Error:
APA Statement:
Report z scores to two decimals and p values to three decimals. If the p value is less than .001, report it as p < .001.
Problem 3 µ = 100
σ =30
Child Concentration z score p value
1 75
2 81
3 89
4 99
5 115
6 127
7 138
8 139
9 142
10 148
Add your 10 APA-style conclusion statements for Problem 3 here.
Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?
________________________________________
PROBLEM #4
Problem 4: µ = 100 ; σ =40
Child Concentration Z P Hypothesis Error
Child 1 75 z= p= Decision about H0: Possible Error:
APA Statement:
Child 2 81 z= p= Decision about H0: Possible Error:
APA Statement:
Child 3 89 z= p= Decision about H0: Possible Error:
APA Statement:
Child 4 99 z= p= Decision about H0: Possible Error:
APA Statement:
Child 5 115 z= p= Decision about H0: Possible Error:
APA Statement:
Child 6 127 z= p= Decision about H0: Possible Error:
APA Statement:
Child 7 138 z= p= Decision about H0: Possible Error:
APA Statement:
Child 8 139 z= p= Decision about H0: Possible Error:
APA Statement:
Child 9 142 z= p= Decision about H0: Possible Error:
APA Statement:
Child 10 148 z= p= Decision about H0: Possible Error:
APA Statement:
Report z scores to two decimals and p values to three decimals. If the p value is less than .001, report it as p < .001.
Problem 4 µ = 100
σ =40
Child Concentration z score p value
1 75
2 81
3 89
4 99
5 115
6 127
7 138
8 139
9 142
10 148
Add your 10 APA-style conclusion statements for Problem 4 here.
Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?
What happens to the “significance” of each child’s data as the data are progressively more dispersed?
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