The Hypothesis Operator
The hypothesis operator requires a summary function as its left operand and a relational function as its right operand. The left argument is a sample vector, and the right argument is a hypothesized value or another sample vector.
The left argument is the confidence level; it is optional--the default is a 95%. The left operand is a summary function, typically: mean, proportion, variance or sdev. The right operand can be either a numeric vector, a 2-column matrix representing a frequency distribution, or a namespace containing count, mean and variance.
The syntax of the hypothesis operator is:
[ConfLevel] report SampleVector summaryFunction hypothesis relationalFunction Value|SampleVector2
Some examples follow:
A sample of 15 heights of college students was taken and is listed below:
Height←69 69 63 63 66 67 73 67 70 72 71 63 68 63 66
Test whether the mean height is 68 inches:
report Height mean hypothesis = 68
────────────────────────────────────────────────
_
X =67.33333
s =3.37357
n =15
Standard Error: 0.87105
Hypothesis Test
H₀: µ=68 (Claim) H₁: µ≠68
µ
┌────────────────────────┬───────────────────┐
│Test Statistic: │P-Value: │
│t=0.7653587777 │p=0.45677 │
├────────────────────────┼───────────────────┤
│Critical Value: │Significance Level:│
│t(α/2;df=14)=2.144786686│α=0.05 │
└────────────────────────┴───────────────────┘
Conclusion: Fail to reject H₀
────────────────────────────────────────────────
Test at the 10% significance level whether the proportion of students taller than 70 inches is greater than 9%.
0.1 report (Height > 70) proportion hypothesis proportion > .09
────────────────────────────────────────
^
p =0.20000
n =15
Standard Error: 0.07389
Hypothesis Test
H₀: p≤0.09 H₁: p>0.09 (Claim)
┌────────────────┬───────────────────┐
│Test Statistic: │P-Value: │
│Z=1.488662895 │p=0.06829 │
├────────────────┼───────────────────┤
│Critical Value: │Significance Level:│
│Z(α)=1.281551837│α=0.1 │
└────────────────┴───────────────────┘
Conclusion: Reject H₀
────────────────────────────────────────
Test at 1% confidence whether the standard deviation of heights is greater than 3 inches.
0.01 report Height sdev hypothesis > 3
───────────────────────────────────────────────
s =3.37357
n =15
Hypothesis Test
H₀: σ≤3 H₁: σ>3 (Claim)
┌───────────────────────┬───────────────────┐
│Test Statistic: │P-Value: │
│χ²=17.7037037 │p=0.22061 │
├───────────────────────┼───────────────────┤
│Critical Value: │Significance Level:│
│χ²(α;df=14)=29.14123773│α=0.01 │
└───────────────────────┴───────────────────┘
Conclusion: Fail to reject H₀
───────────────────────────────────────────────
Find an 80% confidence interval for the median.
0.8 median confInt Height
66 69
A sample of 85 home sales from a certain city was taken and the average price was $149,000 with a standard devation of $25,000. Test the hypothesis that the average home price is less than $155,000.
report (stats 85 149000 25000) mean hypothesis < 155000
─────────────────────────────────────────────
_
X =149000
s =25000
n =85
Standard Error: 2711.63072
Hypothesis Test
H₀: µ≥155000 H₁: µ<155000 (Claim)
┌─────────────────────┬───────────────────┐
│Test Statistic: │P-Value: │
│t=2.21269067 │p=0.01482 │
├─────────────────────┼───────────────────┤
│Critical Value: │Significance Level:│
│t(α;df=84)=1.66319669│α=0.05 │
└─────────────────────┴───────────────────┘
Conclusion: Reject H₀
─────────────────────────────────────────────
A sample of 38 students were asked how many siblings each had. 2 were only children, 17 had one sibiling, 11 had two siblings, 7 and three and one had four. Test whether the average student has at least one brother or sister.
FD←0 1 2 3 4,[1.5]2 17 11 7 1
0.10 report FD mean hypothesis ge 1
────────────────────────────────────────────
_
X =1.68421
s =0.93304
n =5
Standard Error: 0.41727
Hypothesis Test
H₀: µ≥1 (Claim) H₁: µ<1
┌────────────────────┬───────────────────┐
│Test Statistic: │P-Value: │
│t=¯1.639746471 │p=0.91180 │
├────────────────────┼───────────────────┤
│Critical Value: │Significance Level:│
│t(α;df=4)=1.53320625│α=0.1 │
└────────────────────┴───────────────────┘
Conclusion: Fail to reject H₀
────────────────────────────────────────────