Looking at Confidence Intervals

Transcription

1 1 of 40 12/26/2015 9:55 PM Return to the Set Up Look at Intervals page This page extends the illustration of taking samples from a population seen in the page lookatsamples.htm. As such we will start with a population. The goal characteristics of that popultion are: Kind=Normal Intended mean = 500 Intended st. dev.= to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to

2 2 of 40 12/26/2015 9:55 PM 340 to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to

3 3 of 40 12/26/2015 9:55 PM 770 to to to to to to to to to to to to to to to to to to to to to to to After generating the population we find that Achieved mean = Achieved st. dev.= Now we will sort the values and display them again: 0 to to to to to to to to to to to to to to

4 4 of 40 12/26/2015 9:55 PM 140 to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to

5 5 of 40 12/26/2015 9:55 PM 570 to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to to

6 6 of 40 12/26/2015 9:55 PM Name Interval Freq Name Interval Freq Name Interval Freq Name Interval Freq A [168.00,196.00) 1 G [336.00,364.00) 36 M [504.00,532.00) 108 T [672.00,700.00) 17 B [196.00,224.00) 3 H [364.00,392.00) 47 N [532.00,560.00) 93 U [700.00,728.00) 13 C [224.00,252.00) 3 I [392.00,420.00) 78 P [560.00,588.00) 81 V [728.00,756.00) 6 D [252.00,280.00) 10 J [420.00,448.00) 93 Q [588.00,616.00) 66 W [756.00,784.00) 4 E [280.00,308.00) 8 K [448.00,476.00) 108 R [616.00,644.00) 48 X [784.00,812.00) 2 F [308.00,336.00) 21 L [476.00,504.00) 120 S [644.00,672.00) 32 Y [812.00,840.00) 2 G [336.00,364.00) 36 M [504.00,532.00) 108 T [672.00,700.00) 17 For starters, we will produce a histogram of the data distribution [ , ): X [ , ): [ , ): [ , ): X [ , ): [ , ): X

7 7 of 40 12/26/2015 9:55 PM [ , ): [ , ): X [ , ): [ , ): XX [ , ): [ , ): X [ , ): XXX [ , ): X [ , ): XXXX [ , ): XX [ , ): X [ , ): XX [ , ): XXX [ , ): X [ , ): XXXX [ , ): XXX [ , ): XXXXXXXXX [ , ): XX [ , ): XXXXXXXX [ , ): XXXXXXXXX [ , ): XXXXXXXX [ , ): XXXXXXXXX [ , ): XXXXXXXXXXX [ , ): XXXXXXXX [ , ): XXXXXXXXXX [ , ): XXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXX [ , ): XXXXXXXXXXXXXX [ , ): XXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXX

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9 9 of 40 12/26/2015 9:55 PM [ , ) : X [ , ) : X Now we will embark on drawing 1000 samples of size 40 and from each such sample we will get a sample mean and a sample standard deviation, both of which we will record. Here is a display of the 1000 sample means (top value) and standard deviations (bottom value): sample number 0 to 9 10 to to to to to to to to to to to to to to to to to to to 199 mean std. dev mean std. dev mean std. dev mean std. dev mean std. dev mean std. dev mean std. dev mean std. dev mean std. dev mean std. dev

10 10 of 40 12/26/2015 9:55 PM 200 to to to to to to to to to to to to to to to to to to to to to to to to

11 11 of 40 12/26/2015 9:55 PM 440 to to to to to to to to to to to to to to to to to to to to to to to to

12 12 of 40 12/26/2015 9:55 PM 680 to to to to to to to to to to to to to to to to to to to to to to to to

13 13 of 40 12/26/2015 9:55 PM 920 to to to to to to to to Having taken those samples we can recall or compute the Achieved mean of data = Mean of the mean values = Standard deviation of the mean values = Predicted standard deviation of means = Achieved standard deviation of data = Mean of the standard deviation values = Standard deviation of the standard deviation values =

14 14 of 40 12/26/2015 9:55 PM Name Interval Freq Name Interval Freq Name Interval Freq Name Interval Freq A [440.00,444.00) 1 H [468.00,472.00) 10 P [496.00,500.00) 99 W [524.00,528.00) 32 B [444.00,448.00) 0 I [472.00,476.00) 28 Q [500.00,504.00) 85 X [528.00,532.00) 13 C [448.00,452.00) 1 J [476.00,480.00) 42 R [504.00,508.00) 102 Y [532.00,536.00) 16 D [452.00,456.00) 1 K [480.00,484.00) 45 S [508.00,512.00) 88 Z [536.00,540.00) 0 E [456.00,460.00) 4 L [484.00,488.00) 73 T [512.00,516.00) 54 a [540.00,544.00) 2 F [460.00,464.00) 7 M [488.00,492.00) 90 U [516.00,520.00) 57 b [544.00,548.00) 3 G [464.00,468.00) 12 N [492.00,496.00) 96 V [520.00,524.00) 37 c [548.00,552.00) 1 H [468.00,472.00) 10 P [496.00,500.00) 99 W [524.00,528.00) 32 d [552.00,556.00) 1 The lowest sample mean was The highest sample mean was A histogram of the sample means: [ , ) : [ , ) : X [ , ) : [ , ) :

15 15 of 40 12/26/2015 9:55 PM [ , ): [ , ): [ , ): [ , ): X [ , ): [ , ): X [ , ): XX [ , ): [ , ): XX [ , ): [ , ): XXXXX [ , ): XXXX [ , ): XXXXXXX [ , ): XXX [ , ): XXX [ , ): XXXXX [ , ): XXXXXX [ , ): XXXXXXXXXXXXXX [ , ): XXXXXXXXX [ , ): XXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXXXX

16 16 of 40 12/26/2015 9:55 PM [ , ): XXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXXXXXXXXX [ , ): XXXXXXXXXXXXXX [ , ): XXXXXXXX [ , ): XXXXX [ , ): XXXXXXX [ , ): XXX [ , ): XXXXXXX [ , ): XXXXXXX [ , ): XX [ , ): [ , ): [ , ): XX [ , ): [ , ): [ , ): XX [ , ): X [ , ): X [ , ): [ , ): [ , ): [ , ): X [ , ): Now we will use those 1000 pairs of values, the mean and standard deviation of each of the 1000 samples, to generate 1000 confidence intervals. Actually, we will generate two confidence intervals for each sample: a confidence interval if we know the standard deviation of the underlying population (which we know is ) and a confidence interval if we do not know that value. We will be creating 90.00% confidence intervals. Note that the z-value for is , and the t-value for 39 degrees of freedom is Also, for the known population standard deviation case, the standard error of the mean will always be that value, namely, divided by the square root of the sample size, in this case that is the square root of 40, or , which means that when we know the population standard deviation then the margin of error will always be For confidence intervals where we do not know the population standard deviation we use the sample standard deviation divided by the square root of the sample size. Thus, in those cases, the margin of error needs to be recomputed for every sample. This is shown in the table below. Sample Number Sample mean MOE Known Population Standard Deviation Confidence Interval mean in? Sample St. Dev. Unknown population Standard deviation St. Err of mean MOE Confidence Interval ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) Mean in?

17 17 of 40 12/26/2015 9:55 PM ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

18 18 of 40 12/26/2015 9:55 PM ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

19 19 of 40 12/26/2015 9:55 PM ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) FAIL ( , ) ( , ) ( , ) FAIL ( , ) ( , ) ( , ) ( , ) FAIL ( , ) FAIL ( , ) FAIL ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) FAIL ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )