Week 4 - Homework 4

All of these inferential statistics analyses are simply to determine how closely our SAMPLE data reflect the TRUE POPULATION. Closeness is defined as CONFIDENCE: typically 95%, but 90% and 99% (medical) are common as well.

UNUSUAL data values are now defined as having a specified LOW percentage chance of occurring (typically 5% chance or less if the confidence level is 95%). Calculated TEST STATISTICS than have a defined LOW probability of occurring result in our REJECTING our hypothesis.

Problem (1):Fill in the Frequency Table for this 100 value data setAND calculate the MEAN, Variance, and Standard Deviation.

1	11	13	15	20	21	21	25	26	40
38	38	38	39	39	39	40	40	48	60
46	46	47	47	47	47	48	48	69	70
54	54	55	55	55	55	56	57	71	74
26	28	28	30	30	31	32	32	76	79
41	41	41	41	42	43	43	43	90	59
49	49	49	50	50	51	51	51	51	52
61	61	63	63	64	65	66	67	52	53
32	33	35	36	36	37	38	26	53	53
43	43	44	45	45	46	46	40	53	48

Groups (ranges)	Frequency	Relative Frequency	Cumulative Relative Freq	CRF as a %
1 - 10	1	.01	.01	.1
11 - 20	4	.04	.05
21 - 30	10	.10	.15
31 - 40	20	.20	.35
41 - 50	30	.30	.65
51 - 60	20	.20	.85
61 - 70	10	.10	.95
71 - 80	4	.04	.99	.99%
81 - 90	1	.01	1	100%
Total

2) Create a Histogram using the Frequency column. Is it SKEWED and if so + or - ?

3) Using the Cumulative Relative frequency column (I would suggest) find the PROBABILITIES of data being:

1. Less or equal to 60 ( <60)
2. Less than 61 (< 61)
3. Greater than 60
4. Less than or equal to 20 ( < 20)
5. Less than or equal to 60, but 21 or greater (between 21 and 60)?
6. Greater than 30
7. Greater than 40 but less than 81
8. Between51 and 80 inclusive

4) Do ANY of the Relative Frequency rows have probabilities LESS THAN 0.05 or 5% ? Data values in these ranges could be considered RARE or UNUSUAL. Which data values are these?

5) Using the attached z-Tables find the PROBABILITIES of data being to the LEFT of these + z-value standard deviations

-z-value	Probability to LEFT	Probability to RIGHT	+z-value	Probability to LEFT	Probability to Right
(a) z= -0.11 (b) z = -1.30 (c) z = -3.2			(d) z = +3.1 (e) z = + 1.65 (f) z = +2.57

6) How many standard deviations to the LEFT (-z-value) correspond to these probabilities

(a) 0.0329 ; (b)0.0007 ; (c) 0.3859

7) How many standard deviations to the RIGHT (+z-value) correspond to these probabilities (to the left)

(a) 0.9995 ; (b) 0.9864 ; (c) 0.6103

8) (Trickier) What number of standard deviations ( z-value) corresponds to these percentages (probabilities) of data being ABOVE (to the right)

(a) 0.119 ; (b) 0.0495 ; (c) 0.409

9) Let's say a data value has a standardized z-value of 0.0446, which is less than 5% (0.0500).We want to know what the actual x-value is that has this standardized z-value. How do we back-calculate it from our original z-formula of z = (x-mean)/SD? Dust off you algebra and convert his formula to x = ______________

Now, what is the z-value that has a probability of 0.0446? Then, if the mean of the original x-value data set is 45 and the SD is 15, what is the x-value in question?

10) SHOW YOUR WORK(Data made up) x = blood pressure of a person in Maryland measured in mm Hg (Hg = mercury). We MUST assume these blood pressures have a NORMAL Distribution.

We KNOW the true Populationmean pressure is 125 mm Hg ( = 125) and Population std dev (SD) is 20 mm Hg ( = 20), which means we don't need to rely on sample data.

(a) Probability a blood pressure is GREATER THAN or equal to 140 mm Hg (P(x) 140 mm Hg)

1) Standardize x = 140 to get the z-value which represents how many standard deviations this x-value is from the meanusing z = (x - mean)/SD z = ________

2) Depending on whether this z-value is positive or negative go to the appropriate z-Table and find the probability (area to the LEFT) that this z-number of standard deviations represents:_________

NOTE: IF it's an x is GREATER THAN problem (like this one) we want the area (probability) to the RIGHT. But z-values always give areas to the left, so once we have the probability corresponding to a calculated z-value, we SUBTRACT that probability from 1.00 or 100% to get the probability to the right (the x is greater than probability)

3) From the z-Table we now know the probability to the LEFT (which would represent the probability of x being LESS than 140) so we need to subtract that probability from 100% or 1.000 to get the probability to the RIGHT (x being greater than 140). That probability is: ___________

But, if it's an x is LESS THANproblem, we simply get that probability directly from our z-value and the z-Table.

(b) Probability a blood pressure is LESS THAN 150 mm Hg ( P(x) 150 mm Hg) x = 150

Follow same steps as with (a)

Z = ______ , Probability = ________

(c)Probability blood pressure is between 115 and 130 mm HG

Need to calculate TWO probabilities: x < 130 and x < 115 and find the difference (just like you did with Cumulative Relative frequencies for probabilities between two ranges or groups).

(d) Is a blood pressure of 160 mm Hg UNUSUAL? ( x = 160) z = _______; Probability = _______

(e) What blood pressure do 95% of Marylanders have one LESS THAN (not realistic of course)?

What z-value corresponds to a 95% probability to the LEFT? It's about __________

Now, what x-value corresponds to this z-value (remember this formula) ? _____________

11) We do NOT know the true population mean and standard deviation, so we must use the SAMPLE mean and standard deviation. We need to know the sample size as well (represented by "n")

Per the Central Limit Theorem we can assume the Sample mean is the same as the Population mean. However, the Population standard deviation is approximated by dividing the sample SD by the square root of the sample size: Population SD = Sample SD/ SQRT(n)

Work the following for a sample size of n = 11 with a sample mean of 250 and a sample standard deviation of 25.

(a) What is the probability that x < 240 ? z = _______, Probability (x) = _________(did you use the area to the LEFT or the RIGHT?)_________

(b)P(x) if x > 265 ? z-value:_______ ; Probability: ________ (this is the probability (area) to the LEFT, is that what you want here? If not what is the correct probability? __________(How did you calculate it?)

(c) REPEAT (a) and (b) IF the sample size (n) is 40 instead of 11.

What did you use as the mean:___________, what did you use for the new population SD:____________

P(x < 240): z = _________, Probability:____________ (left or right?) UNusual at 5% level?

P(x > 265): z = _________, Probability: ____________ (left or right?) UNusual at 5% level?