목차

• Fundamentals of Hypothesis Testing: One-Sample Tests
• 개요
• INTRO
• Hypothesis Tests for Proportions
• Proportions
• Example: Z Test for Proportion
  • PROBLEM
  • SOLVE 1
  • SOLVE 2: p-value
• 문제

👀, 🤷‍♀️ , 📜, 📝
이 아이콘들을 누르시면 정답, 개념 부가 설명을 보실 수 있습니다:)

Fundamentals of Hypothesis Testing: One-Sample Tests

즉, 가설검증이다.
test라는 위딩이 들어갔다는 것은 이 추정의 결과로 인한 의사결정까지 한다는 것이다.

One-Sample Tests: 데이터 세트가 하나이다.

• One-Sample Tests: 우리나라 사람의 아이큐가 100 이상이다 -> 우리나라 사람의 아이큐라는 하나의 데이터 세트가 필요하다
• Two-Sample Tests: 우리나라와 일본의 아이큐를 비교해보면 우리나라가 더 높다 -> 우리나라 아이쿠 데이터 세트와 일본의 아이큐 데이터세트가 필요하다. 즉, 비교, 변화 검증에 많이 쓰인다.

개요

아래의 도식을 하나씩 자세히 알아볼 것이다.

• σ known: Fundamentals of Hypothesis Testing Methodology
• σ Unknown: t-Test of Hypothesis for the Mean
• One-Tail Tests(도식 전체에 적용)
• Population Proportion: Z-Test of Hypothesis for the Proportion
• Potential Hypothesis Testing Pitfalls and Ethical Issues(추가)

INTRO

[Hypothesis Tests for Proportions]
Involves categorical variables

• Two possible outcomes
• Possesses characteristic of interest
• Does not possess characteristic of interest
• Fraction or proportion of the population in the category of interest is denoted by π
• 📌 즉 확률(π)을 다루는 것이다.

Hypothesis Tests for Proportions

Proportions

[Sample proportion]

• in the category of interest is denoted by p
  

[조건]

• 📌 정규성 확립: When both nπ and n(1-π) are at least 5,
• p can be approximated by a normal distribution with mean and standard deviation
  

Example: Z Test for Proportion

Proportion은 무조건 z-table을 쓴다!

• z-table 더 알아보기
• t-table 더 알아보기

PROBLEM

A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the a = 0.05 significance level.

➡️ it receives 8% responses: 퍼센트가 나왔으므로 proprotion 문제라는 것을 알 수 있다.

SOLVE 1

1) Check: 정규성 확인

• nπ = (500)(.08) = 40
• n(1-π) = (500)(.92) = 460

2) 조건 확인

• \(H_0\): π = 0.08
• \(H_1\): π != 0.08
• a = 0.05
• n = 500, p = 0.05

3) Find Rejection Region: Critical Values
a를 이용하여 z-table에서 구하기
더 알아보기

4) Test Statistic

3) Decision
Reach a decision and interpret the result:
➡️ Reject \(H_0\) at a = 0.05

• There is sufficient evidence to reject the company’s claim of 8% response rate

SOLVE 2: p-value

p-value 란?

Calculate the p-value and compare to a

➡️ Reject H0 since p-value = 0.0136 < a = 0.

• There is sufficient evidence to reject the company’s claim of 8% response rate

문제

[TABLE 9-1]
An appliance manufacturer claims to have developed a compact microwave oven that consumesan average of no more than 250 W. From previous studies, it is believed that power consumptionfor microwave ovens is normally distributed with a standard deviation of 15 W. A consumergroup has decided to try to discover if the claim appears true. They take a sample of 20 microwave ovens and find that they consume an average of 257.3 W.

1. Referring to Table 9-1, the parameter of interest is
a) the mean power consumption of the 20 microwave ovens.
b) the mean power consumption of all such microwave ovens.
c) 250
d) 257.3

2. Referring to Table 9-1, the appropriate hypotheses to determine if the manufacturer’s claim appears reasonable are:
a) \(𝐻_0\): 𝜇 = 250 versus \(𝐻_1\): 𝜇 ≠ 250
b) \(𝐻_0\): 𝜇 ≥ 250 versus \(𝐻_1\): 𝜇 < 250
c) \(𝐻_0\): 𝜇 ≤ 250 versus \(𝐻_1\): 𝜇 > 250
d) \(𝐻_0\): 𝜇 ≥ 257.3 versus \(𝐻_1\): 𝜇 < 257.3

3. Referring to Table 9-1, for a test with a level of significance of 0.05, the critical value would be ________.

4. Referring to Table 9.1, the value of the test statistic is ________.

5. Referring to Table 9-1, the p-value of the test is ________.

6. True or False: Referring to Table 9-1, for this test to be valid, it is necessary that the power consumption for microwave ovens has a normal distribution.

7. True or False: Referring to Table 9-1, the null hypothesis will be rejected at 5% level of significance.

8. True or False: Referring to Table 9-3, the null hypothesis will be rejected at 1% level of significance.

[TABLE 9-3] How many Kleenex should the Kimberly Clark Corporation package of tissues contain? Researchers determined that 60 tissues is the average number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: X = 52, s = 22.

9. Referring to Table 9-3, give the null and alternative hypotheses to determine if the number of tissues used during a cold is less than 60.
A. \(𝐻_0\): 𝜇 ≤ 60 and \(𝐻_1\): 𝜇 > 60.
B. \(𝐻_0\): 𝜇 ≥ 60 and \(𝐻_1\): 𝜇 < 60.
C. \(𝐻_0\): x ̅ ≥ 60 and \(𝐻_1\): x ̅ < 60.
D. \(𝐻_0\): x ̅ = 52 and \(𝐻_1\): x ̅ != 52.

10. Referring to Table 9-3, using the sample information provided, calculate the value of the test statistic.
A. 𝑡 = (52 − 60)/22
B. 𝑡 = (52 − 60)/(22/100)
C. 𝑡 = (52 − 60)/(22/\(100^2\))
D. 𝑡 = (52 − 60)/(22/10)

11. Referring to Table 9-3, suppose the alternative we wanted to test was \(𝐻_1\): 𝜇 < 60 . State the correct rejection region for a = 0.05.
A. Reject \(𝐻_0\) if t > 1.6604.
B. Reject \(𝐻_0\) if t < – 1.6604.
C. Reject \(𝐻_0\) if t > 1.9842 or Z < – 1.9842.
D. Reject \(𝐻_0\) if t < – 1.9842.

12. Referring to Table 9-3, suppose the test statistic does fall in the rejection region at a = 0.05. Which of the following decision is correct?
A. At a = 0.05, we do not reject \(𝐻_0\).
B. At a = 0.05, we reject \(𝐻_0\).
C. At a = 0.05, we accept \(𝐻_0\).
D. At a = 0.10, we do not reject \(𝐻_0\).

13. Referring to Table 9-3, suppose the test statistic does fall in the rejection region at a = 0.05. Which of the following conclusion is correct?
A. At a = 0.05, there is not sufficient evidence to conclude that the average number of tissues used during a cold is 60 tissues.
B. At a = 0.05, there is sufficient evidence to conclude that the average number of tissues used during a cold is 60 tissues.
C. At a = 0.05, there is not sufficient evidence to conclude that the average number of tissues used during a cold is not 60 tissues.
D. At a = 0.05, there is sufficient evidence to conclude that the average number of tissues used during a cold is not 60 tissues.

[TABLE 9-5] A major videocassette rental chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in the area are equipped with videocassette recorders (VCRs). It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have VCRs. 14. Referring to Table 9-5, state the test of interest to the rental chain.
A. \(𝐻_0\): 𝜋 ≤ 0.32 versus \(𝐻_1\): 𝜋 > 0.32
B. \(𝐻_0\): 𝜋 ≤ 0.25 versus \(𝐻_1\): 𝜋 > 0.25
C. \(𝐻_0\): 𝜋 ≤ 5,000 versus \(𝐻_1\): 𝜋 > 5,000
D. \(𝐻_0\): 𝜇 ≤ 5,000 and \(𝐻_1\): 𝜇 > 5,000

15. Referring to Table 9-5, the value of the test statistic in this problem is approximately equal to:
A. 2.80
B. 2.60
C. 1.94
D. 1.30

16. Referring to Table 9-5, the p-value associated with the test statistic in this problem is approximately equal to:
A. 0.0100
B. 0.0051
C. 0.0026
D. 0.0013

17. Referring to Table 9-5, the decision on the hypothesis test using a 3% level of significance is:
A. to reject \(𝐻_0\) in favor of \(𝐻_1\).
B. to accept \(𝐻_0\) in favor of \(𝐻_1\).
C. to fail to reject \(𝐻_0\) in favor of \(𝐻_1\).
D. we cannot tell what the decision should be from the information given.

18. Referring to Table 9-5, the rental chain’s conclusion from the hypothesis test using a 3% level of significance is:
A. to open a new store.
B. not to open a new store.
C. to delay opening a new store until additional evidence is collected.
D. we cannot tell what the decision should be from the information given.