please see attachment

In this unit, you will investigate the normal probability curve (the bell curve). Many variables, such as height and weight are “normally distributed.” This means, for example, that if you were to collect 10,000 female adult human heights, the histogram of that data would be shaped like a “bell” (with “most” of the data near the center or mean).

Use the following z table portion to assist you with answering the Discussion topics. There is a full z table in Course Resources.

Different university departments use different tests to compare student performance and to determine graduate admission status. Three such tests are the GMAT, the LSAT, and the GRE.

1. Across the USA, results for these exams are normally distributed. What does that mean and why is this the case?

2. If you were to create a histogram of all GRE scores, what would you expect the histogram to look like? Would it be symmetrical? Would it be bell shaped? How many modes would it likely have? Would it be skewed?

3. Suppose that the mean GRE score for the USA is 500 and the standard deviation is 75. Use the 68-95-99.7 (Empirical) Rule to determine the percentage of students likely to get a score between 350 and 650? What percentage of students will get a score above 500? What percentage of students will get a score below 275? Is a score below 275 significantly different from the mean? Why or why not?

4. Choose any GRE score between 200 and 800. Be sure that you do not choose a score that a fellow student has already selected. Using your chosen score, how many standard deviations from the mean is your score? (This value is called the z-value). Using the table above (or the z table in Course Resources), what percentage of students will likely get a score below this value? What percentage of students is likely to get a score above this value?

Hints: The “standard score,” the “z score,” the “z value,” and the “number of standard deviations from the mean” are all saying the same thing. If you cannot find your exact score on the table, use the closest value or use the z table in Course Resources. There is a tutorial that can assist located in Course Resources.

Standard Normal (

z

) Table

0) =

0

62

z 0.00

0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003

0.0005 0.0005

0.0004 0.0004 0.0004 0.0004 0.0004 0.0003

0.0007

0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005

0.0009 0.0009

0.0008 0.0008 0.0008 0.0007 0.0007

0.0013 0.0013

0.0012

0.0011 0.0011 0.0010 0.0010

0.0018

0.0016

0.0015

0.0014

0.0023

0.0021

0.0019

0.0026

-2.5

151

119

085

0.0 0.5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

23

0.9

1.0

1.1

1.2

15

1.3

1.4

1.5

1.6

05

1.7

1.8

1.9

2.0

2.1

2.2

2.3

01

2.4

2.5

2.6

2.7

2.8 0.9974

0.9977

0.9979

2.9 0.9981

0.9982

0.9984

0.9985

0.9986

3.0

0.9987 0.9987

0.9988

0.9989 0.9989

0.9990

3.1 0.9990

0.9991 0.9991

0.9992 0.9992 0.9992

0.9993

3.2 0.9993 0.9993

0.9994 0.9994 0.9994 0.9994

0.9995 0.9995

3.3 0.9995 0.9995 0.9995

0.9996 0.9996 0.9996 0.9996 0.9996

3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997

																									Standard Normal Distribution Table (z table)
Table is written for the probabilities to the left of z:
Ex: P(z<	–	2.5		0.0
0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
–	3.4										0.0003	0.0002
–	3.3						0.0005						0.0004
–	3.2				0.0007					0.0006
–	3.1			0.0010			0.0009				0.0008
–	3.0			0.0013		0.0012			0.0011
–	2.9		0.0019		0.0018	0.0017		0.0016		0.0015		0.0014
–	2.8		0.0026	0.0025	0.0024		0.0023	0.0022		0.0021	0.0020
–	2.7	0.0035	0.0034	0.0033	0.0032	0.0031	0.0030	0.0029	0.0028	0.0027
–	2.6	0.0047	0.0045	0.0044	0.0043	0.0041	0.0040	0.0039	0.0038	0.0037	0.0036
0.0062	0.0060	0.0059	0.0057	0.0055	0.0054	0.0052	0.0051	0.0049	0.0048
–	2.4	0.0082	0.0080	0.0078	0.0075	0.0073	0.0071	0.0069	0.0068	0.0066	0.0064
–	2.3	0.0107	0.0104	0.0102	0.0099	0.0096	0.0094	0.0091	0.0089	0.0087	0.0084
–	2.2	0.0139	0.0136	0.0132	0.0129	0.0125	0.0122	0.0119	0.0116	0.0113	0.0110
–	2.1	0.0179	0.0174	0.0170	0.0166	0.0162	0.0158	0.0154	0.0150	0.0146	0.0143
–	2.0	0.0228	0.0222	0.0217	0.0212	0.0207	0.0202	0.0197	0.0192	0.0188	0.0183
–	1.9	0.0287	0.0281	0.0274	0.0268	0.0262	0.0256	0.0250	0.0244	0.0239	0.0233
–	1.8	0.0359	0.0351	0.0344	0.0336	0.0329	0.0322	0.0314	0.0307	0.0301	0.0294
–	1.7	0.0446	0.0436	0.0427	0.0418	0.0409	0.0401	0.0392	0.0384	0.0375	0.0367
–	1.6	0.0548	0.0537	0.0526	0.0516	0.0505	0.0495	0.0485	0.0475	0.0465	0.0455
–	1.5	0.0668	0.0655	0.0643	0.0630	0.0618	0.0606	0.0594	0.0582	0.0571	0.0559
–	1.4	0.0808	0.0793	0.0778	0.0764	0.0749	0.0735	0.0721	0.0708	0.0694	0.0681
–	1.3	0.0968	0.0951	0.0934	0.0918	0.0901	0.0885	0.0869	0.0853	0.0838	0.0823
–	1.2		0.1	0.1131	0.1112	0.1093	0.1075	0.1056	0.1038	0.1020	0.1003	0.0985
–	1.1	0.1357	0.1335	0.1314	0.1292	0.1271	0.1251	0.1230	0.1210	0.1190	0.1170
–	1.0	0.1587	0.1562	0.1539	0.1515	0.1492	0.1469	0.1446	0.1423	0.1401	0.1379
–	0.9	0.1841	0.1814	0.1788	0.1762	0.1736	0.1711	0.1685	0.1660	0.1635	0.1611
–	0.8		0.2	0.2090	0.2061	0.2033	0.2005	0.1977	0.1949	0.1922	0.1894	0.1867
–	0.7	0.2420	0.2389	0.2358	0.2327	0.2296	0.2266	0.2236	0.2206	0.2177	0.2148
–	0.6	0.2743	0.2709	0.2676	0.2643	0.2611	0.2578	0.2546	0.2514	0.2483	0.2451
–	0.5		0.3	0.3050	0.3015	0.2981	0.2946	0.2912	0.2877	0.2843	0.2810	0.2776
–	0.4	0.3446	0.3409	0.3372	0.3336	0.3300	0.3264	0.3228	0.3192	0.3156	0.3121
-0.3	0.3821	0.3783	0.3745	0.3707	0.3669	0.3632	0.3594	0.3557	0.3520	0.3483
-0.2	0.4207	0.4168	0.4129	0.4090	0.4052	0.4013	0.3974	0.3936	0.3897	0.3859
-0.1	0.4602	0.4562	0.4522	0.4483	0.4443	0.4404	0.4364	0.4325	0.4286	0.4247
-0.0		0.5000	0.4960	0.4920	0.4880	0.4840	0.4801	0.4761	0.4721	0.4681	0.4641
0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.6915	0.5960	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7220
0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.7881	0.7910	0.7939	0.7967	0.7995		0.80	0.8051	0.8078	0.8106	0.8133
0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997		0.90
0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
0.9452	0.9463	0.9474	0.9484	0.9495		0.95	0.9515	0.9525	0.9535	0.9545
0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
0.9893	0.9896	0.9898		0.99	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973		0.9974
0.9975	0.9976		0.9977	0.9978		0.9979	0.9980		0.9981
	0.9982	0.9983		0.9984		0.9985		0.9986
		0.9987		0.9988			0.9989			0.9990
		0.9991				0.9992				0.9993
				0.9994						0.9995
					0.9996										0.9997
0.9998

z

Critical Values List

0.80

0.90

0.95

0.99

1.960 1.645

One tail test *

Two tailed test

Critical Values for the Standard Normal Distribution
Confidence Intervals:
Level of Confidence c	zc
1.280
	1.645
	1.960
2.575
Hypothesis Testing (zc):
left-tailed test (one tail)	right-tailed test (one tail)
two tailed test
α = 0.005	α = 0.01	α = 0.02	α = 0.025	α = 0.05
	One tail test *	2.576	2.326
	Two tailed test	± 2.576	± 2.326	± 1.96
α = 0.10	α = 0.20	α = 0.25	α = 0.50
1.282	0.674
± 1.645	± 1.282	± 0.674
* zc will be positive or negative depending on the direction of the tail

Choose any two classmates and review their main posts.

1. Review the student post and evaluate their solutions for using the 68-95-99.7 (Empirical) Rule to determine the percentage of GRE scores between 350 and 650. Are the student’s calculations correct? If yes, note this and if not correct them with an example. Next, explain to the student why 50% of the scores are above 500 and why 50% are below (approximately).

2. Review the student’s GRE score choice from number 4 above. Are the student calculations correct? Include the student’s calculations in your response and note any issues if discovered. Then, offer the student a second example using any other value between 300 and 500. Be sure to explain all the steps in your example to the student and to show all work.

Classmate 1 Sollars

· Across the USA, results for these exams are normally distributed. What does that mean and why is this the case?

First, it should be understood that there is a hard limit to high and low scores. No matter how smart a person is, there can’t be an extreme outlier that could get, say, 5000 on an SAT. Additionally, since a measure such as “competence” is a pretty even variable throughout a population, a normal distribution of scores should be expected if a test is properly written. So, most people score around the middle of the chart, with a few people on either end getting high or low scores. Perfect scores should be as common as perfectly low scores.

· If you were to create a histogram of all GRE scores, what would you expect the histogram to look like? Would it be symmetrical? Would it be bell shaped? How many modes would it likely have? Would it be skewed?

I would expect a histogram of GRE scores to be a unimodal, bell-shaped curve. In all likelihood, it would be symmetric, with as many expected outliers on the high side as on the low side. The mean of the scores would likely occur at the peak of the shape, or very close to it.

· Suppose that the mean GRE score for the USA is 500 and the standard deviation is 75. Use the 68-95-99.7 (Empirical) Rule to determine the percentage of students likely to get a score between 350 and 650? What percentage of students will get a score above 500? What percentage of students will get a score below 275? Is a score below 275 significantly different from the mean? Why or why not?

I’ll use a visual aid for ease of reading:

If the mean GRE score was 500, the following would be true:

Scores between 350 and 650 would be those between -2 and +2 standard deviations from the mean, which would leave us with 95% of people scoring within that range.

Fully 50% of students would be expected to have scores above 500.

A score of 275 is 3 full deviations below the mean, which would give us .15% of students. This is significantly different than the mean, and anyone with that score would be considered an outlier because of how few would be expected to receive the same score or lower.

· Choose any GRE score between 200 and 800. Be sure that you do not choose a score that a fellow student has already selected. Using your chosen score, how many standard deviations from the mean is your score? (This value is called the z-value). Using the table above (or the z table in Course Resources), what percentage of students will likely get a score below this value? What percentage of students is likely to get a score above this value?

I’ll chose a value of 590 for my score. This is 1.200 standard deviations higher than the mean. According to the z-table, 88.49% of students will score lower, which means 11.51% would be expected to score higher.

Classmate 2 (Cummin)

Across the USA, results for these exams are normally distributed. What does that mean and why is this the case?

normally distributed is relating to a bell shape curve. When the data is placed in a graph the information forms the shape of a bell, the mean will be in the middle of the bell and will be the highest amount of that particular variable. 

If you were to create a histogram of all GRE scores, what would you expect the histogram to look like? Would it be symmetrical? Would it be bell shaped? How many modes would it likely have? Would it be skewed?

A histogram of all of the GRE scores would be a bell shaped histogram, and it would be symmetrical. It would be unimodel, meaning that it only has one hump and it wouldn’t be skewed to one side or the other.

Suppose that the mean GRE score for the USA is 500 and the standard deviation is 75. Use the 68-95-99.7 (Empirical) Rule to determine the percentage of students likely to get a score between 350 and 650? What percentage of students will get a score above 500? What percentage of students will get a score below 275? Is a score below 275 significantly different from the mean? Why or why not?

The percentage of scores that would fall between 350-650 would be 95%.

The percentage of scores that would be above 500 would be 50%

The percentage of scores that would be below 275 would be .15%

A score that is below 275 would be severely significant from the mean because it is -3 standard deviations from the mean.

Choose any GRE score between 200 and 800. Be sure that you do not choose a score that a fellow student has already selected. Using your chosen score, how many standard deviations from the mean is your score? (This value is called the z-value). Using the table above (or the z table in Course Resources), what percentage of students will likely get a score below this value? What percentage of students is likely to get a score above this value?

343-500/75 = -2.09.

Using the Standard Normal Distribution table (z table) I see the probability is

0.0188 = 1.88% of scores would be below the value of 343.

Above this value would be 98.12%