A Statistics Project

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HYPOTHESIS TESTING NOTES
2-Tailed Test Tests if a mean or proportion is different from a theoretical value.
Ho: m = 100 Word problem might state that the mean is "equal to" 100.
H1: m <> 100 Word problem might state that the mean is "different from" or "not equal to" 100.
Upper-Tail Test Tests if a mean or proportion is greater than a theoretical value.
Ho: m < 100 Word problem might state that the mean is "no more than" or "at most" 100
H1: m > 100 Word problem might state that the mean is "greater than" 100.
Lower-Tail Test Tests if a mean or proportion is less than a theoretical value.
Ho: m > 100 Word problem might state that the mean is "at least" 100
H1: m < 100 Word problem might state that the mean is "less than" 100.
If the null is rejected, there is sufficient evidence to conclude the alternate hypothesis is true.
If the null is not rejected, there is insufficient evidence to conclude the alternate hypothesis is true.
©2007 DrJimMirabella.com

Hyp Mean - One Sample

One Sample Hypothesis Test for the Mean
Null Hypothesis m = 500
Level of Significance 0.05
Sample Size 30
Sample Mean 535
Standard Deviation 100
Test Statistic (Computed) 1.92
Direction of Test Lower Crit Value Upper Crit Value p-Value Decision
Two-Tailed Test -1.9600 1.9600 0.0552 Do not reject the null hypothesis
H1: m <> 500
Upper-Tail Test n/a 1.6449 0.0276 Reject the null hypothesis
H1: m > 500
Lower-Tail Test -1.6449 n/a 0.9724 Do not reject the null hypothesis
H1: m < 500
If you REJECT the null hypothesis, conclude that H1 is true.
If you DO NOT REJECT the null hypothesis, there is insufficient evidence to conclude that H1 is true.
NEVER conclude that the null hypothesis is true (i.e., we CANNOT ACCEPT the null).
©2007 DrJimMirabella.com
&A
If the sample size is less than 30, the statistic is the t-score. If the sample size is greater than 30, the statistic is the z-score.
This is the hypothesized value that is being tested.
This is the mean of the sample that is being compared to the hypothesized value.
A p-value is the probability of making a type 1 error if you reject the null hypothesis. In other words, it is the probability you would be making a mistake to reject the null.
If the p-value is less than the significance level, than the risk of error is within your tolerance, so you reject the null hypothesis. If the p-value is greater, you do not reject the null because the risk is too great. Another way to look at the decision is that if the test statistic > upper critical value or < lower critical value, the null hypothesis is rejected.
Depending on the alternate hypothesis, you select the appropriate test and make the subsequent decision.
Also referred to as ALPHA. This is your tolerance for error; it is the probability of incorrectly rejecting the null hypothesis.

Hyp Mean - One Sample (data)

Observation # Data One Sample Hypothesis Test for the Mean
Observation 1 1
Observation 2 2 Null Hypothesis m = 20
Observation 3 4 Level of Significance 0.05
Observation 4 5 Sample Size 25
Observation 5 6 Sample Mean 15.96
Observation 6 7 Standard Deviation 10.46
Observation 7 9
Observation 8 9 Test Statistic (Computed) -1.93
Observation 9 10
Observation 10 10
Observation 11 12 Direction of Test Lower Crit Value Upper Crit Value p-Value Decision
Observation 12 12
Observation 13 14 Two-Tailed Test -2.0639 2.0639 0.0653 Do not reject the null hypothesis
Observation 14 15 H1: m <> 20
Observation 15 16
Observation 16 18 Upper-Tail Test n/a 1.7109 0.9673 Do not reject the null hypothesis
Observation 17 20 H1: m > 20
Observation 18 22
Observation 19 23 Lower-Tail Test -1.7109 n/a 0.0327 Reject the null hypothesis
Observation 20 25 H1: m < 20
Observation 21 26
Observation 22 28 If you REJECT the null hypothesis, conclude that H1 is true.
Observation 23 32 If you DO NOT REJECT the null hypothesis, there is insufficient evidence to conclude that H1 is true.
Observation 24 33 NEVER conclude that the null hypothesis is true (i.e., we CANNOT ACCEPT the null).
Observation 25 40
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Observation 29 ©2007 DrJimMirabella.com
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&A
If the sample size is less than 30, the statistic is the t-score. If the sample size is greater than 30, the statistic is the z-score.
This is the hypothesized value that is being tested.
This is the mean of the sample that is being compared to the hypothesized value.
A p-value is the probability of making a type 1 error if you reject the null hypothesis. In other words, it is the probability you would be making a mistake to reject the null.
If the p-value is less than the significance level, than the risk of error is within your tolerance, so you reject the null hypothesis. If the p-value is greater, you do not reject the null because the risk is too great. Another way to look at the decision is that if the test statistic > upper critical value or < lower critical value, the null hypothesis is rejected.
Depending on the alternate hypothesis, you select the appropriate test and make the subsequent decision.
Also referred to as ALPHA. This is your tolerance for error; it is the probability of incorrectly rejecting the null hypothesis.

Hyp Test Prop - One Sample

One Sample Hypothesis Test for the Proportion
Null Hypothesis P = 0.5
Level of Significance 0.05
Number of Successes 59
Sample Size 100
Sample Proportion 0.59 (computed from Successes / Sample Size)
Z Test Statistic (Computed) 1.80
Direction of Test Lower Crit Value Upper Crit Value p-Value Decision
Two-Tailed Test -1.9600 1.9600 0.0719 Do not reject the null hypothesis
H1: P <> 0.5
Upper-Tail Test n/a 1.6449 0.0359 Reject the null hypothesis
H1: P > 0.5
Lower-Tail Test -1.6449 n/a 0.9641 Do not reject the null hypothesis
H1: P < 0.5
If you REJECT the null hypothesis, conclude that H1 is true.
If you DO NOT REJECT the null hypothesis, there is insufficient evidence to conclude that H1 is true.
NEVER conclude that the null hypothesis is true (i.e., we CANNOT ACCEPT the null).
©2007 DrJimMirabella.com
&A
Depending on the alternate hypothesis, you select the appropriate test and make the subsequent decision.
A p-value is the probability of making a type 1 error if you reject the null hypothesis. In other words, it is the probability you would be making a mistake to reject the null.
This is the hypothesized value that is being tested.
Also referred to as ALPHA. This is your tolerance for error; it is the probability of incorrectly rejecting the null hypothesis.
This is the number of observations in the sample that meet the specified condition.
If the p-value is less than the significance level, than the risk of error is within your tolerance, so you reject the null hypothesis. If the p-value is greater, you do not reject the null because the risk is too great. Another way to look at the decision is that if the test statistic > upper critical value or < lower critical value, the null hypothesis is rejected.