Math 2526-03 Applied Stats Spring 2023
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Math 2526-03
Applied Stats
Spring 2023
Hypothesis Testing
Review: Estimation, CI for µ and p.
土 Z* σ
土 t*
^n .
Sample variance
n - 1 .
One of the main purpose in Stats.
Check people’s claim, check the quality of product, etc.
Example 1: The president of this school claims that 15% students can get schol- arships.
To test his claim, we did a survey, checked 100 students, and find only 12 students have scholarship.
Can we reject the claim with confidence 95%?
Example 2: The president of this school claims that average GPA of students in this school is 3.0 with standard deviation 0.3.
But, for our class, with 25 students, we find the class average is 2.95. Can we reject the claim with confidence 90%?
Example 3. The president of this school claims that average GPA of students in this school is 3.0 with standard deviation 0.3. But, for our class, with 25 students, we find the class average is 2.85. Can we reject the claim with confidence 90%?
Basic idea: and µ should be close, and follows a normal distribution.
How to judge and µ are close, or not?
Are x1 and x2 close? d = x1 - x2 = 4, their difference is small?
My annual salary is $57,000. Your annual salary is $59,000. d = $2000. My hourly wage is $15.50. Your hourly wage is $20.60. d = $5.10.
To check whether and µ are close or not, we introduce an important notation in Statistics, that is p-value.
What is p-value?
Then, p-value can be:
can be:
Standardize the process. Change general normal to N(0, 1).
Z = ~ N(0, 1).
If someone claim he knows µ = µ0 , and σ , then, we can find the cutoff point,
- µ0
Z =
we call this as Z-value.
If someone claim he knows µ = µ0 , (but not assume know σ), then, we can
find the cutoff point,
T = - µ0
we call this as T-value.
These are referred to as Test statistic(s).
If p-value< α , Ha , otherwise, H0 .
α is called as significance level.
Example 3. The president of this school claims that average GPA of students in this school is 3.0 with standard deviation 0.3. But, in our class, with 25 students, we find the class average is 2.85. Can we reject the claim with confidence 90%?
Solution.
1. Set hypothesis
H0 : µ = 3.00, Ha : µ < 3.00
2). Calculate test statistic, Z-value
- µ0 2.85 - 3.00
σ/^n 0.3/^25
3). Find p-value.
p = P (Z < -2.5) = 0.006
4). Decision.
Here, α = 0.10, Since p < α, we reject the claim. The average GPA for all students in this school will be less than 3.00.
Example 3’ . The president of this school claims that average GPA of students in this school is 3.0. But, in our class, with 25 students, we find the class average is 2.85, with standard deviation 0.35. Can we reject the claim with confidence 90%?
Solution.
1. Set hypothesis
H0 : µ = 3.00, Ha : µ < 3.00
2). Calculate test statistic, T-value
- µ0 2.85 - 3.00
s/^n 0.35/^25
3). Find p-value.
p = P (T < -2.14)
4). Decision.
Is the p-value less than 0.10?
Example 1: The president of this school claims that 15% students can get schol- arships. To test his claim, we did a survey, checked 100 students, and find only 12
students have scholarship. Can we reject the claim with confidence 95%?
Solution.
1. Set hypothesis
H0 : p = 0.15, Ha : p < 0.15.
2). Calculate test statistic, Z-value
Z = ^pq =
^X
3). Find p-value.
4). Decision.
Hypothesis Testing Process
1). Set hypothesis
µ > µ0 ,
H0 : µ = µ0 ,
Ha : µ µ0
µ < µ0 ,
2). Calculate test statistic, Z-value or T-value
Z = - µ0
T =
^p0 q0 /^n .
3). Determine p-value is small or large? Compare p-value with α, significance level.
4). Make your decision.
If p-value< α , Ha , otherwise, H0 .
Example 4. National norms for a school mathematics proficiency exam are dis- tributed as N(80, 20). A random sample of 60 students from New York City is taken showing a mean proficiency score of 75. Do these sample scores differ significantly from the overall population mean at α = 0.05?
1). Set hypothesis
H0 : µ = 80, Ha : µ < 80
2). Calculate test statistic, Z-value or T-value
- µ0 75 - 80
σ/^n 20/^60
3). Find p-value from Z-Table. pvalue= 0.5 - 0.4738 = 0.0262. 4). Make your decision.
Example 5. According to American health association the average blood pressure of a pregnant women is 120 mm Hg. Collected 15 random samples from pregnant women to check the sample blood pressure is different from accepted standard blood pressure. From the sample, we find = 123, and s = 7.3. At α = 0.10, can we believe those 15 women have significant difference from the others?
1). Set hypothesis
H0 : µ = 120, Ha : µ 120
2). Calculate test statistic, Z-value or T-value
T = = = 1.59.
3). Determine p-value is small or large? Compare p-value with α, check T- table.
4). Make your decision.
Example 6. Will he win the election? To predict the Presidential election, we have done a survey. We checked 2000 people, and find 1050 will vote for Mr. A. Can we predict he will win the election at significance level 0.05? (Use hypothesis testing).
pˆ = = 0.525.
1). Set hypothesis
H0 : p = 0.5, Ha : p > 0.5.
2). Calculate test statistic, Z-value or T-value
Z = = = 2.24.
3). Find p-value.
4). Make your decision.
Example 7. Is this random machine really random? Suppose we let the machine produce 10000 digit numbers, we find there 1090 2’s. Can we believe this random machine is not random at significance level 0.05?
pˆ = 1090/10000 = 0.109.
1). Set hypothesis
H0 : p = 0.1, Ha : p 0.1
2). Calculate test statistic, Z-value or T-value
Z = = = 3.00.
3). Determine p-value is small or large? Compare p-value with α, significance level.
4). Make your decision.
2023-04-10