statistics(Hypothesis testing)

Name: Section:

The examples A1, B1 and C1 are solved. Review them so you get of a round of practice. Then, you need to fill in the empty cells for A2, B2 and C2.

z-test Example A1 Example A2 Example B1 Example B2 Example C1 Example C2

𝐻 𝜇 = 10 𝜇 = 10 𝜇 ≤ 0 𝜇 ≤ 0 𝜇 ≥ 5 𝜇 ≥ 5 𝐻 𝜇 ≠ 10 𝜇 ≠ 10 𝜇 > 0 𝜇 > 0 𝜇 < 5 𝜇 < 5 𝜇 10 10 0 0 5 5

One-sided or two sided

Two-sided Two-sided One-sided One-sided One-sided One-sided

�̅� 15 18 0.1 0.05 4.5 3.5 𝝈 10 10 0.1 0.1 4 4 𝑛 15 15 8 8 20 20

z-statistic 1.936 2.82 -0.559

𝛼 0.01 0.01 0.05 0.05 0.1 0.1

𝛼/2 0.005 0.005 N/A N/A N/A N/A

Critical value (From z table)

2.57 1.645 -1.28

Rejection Region

z>2.57 or

z< -2.57

z>1.64 z<-1.28

Decision

Do not Reject H0 Reject H0 Do not Reject H0

p-value

0.053 0.002 0.288

Degree of evidence against 𝐻

Weak evidence Strong evidence Weak evidence

Fill in the blanks:

Interpretation of Example A1:

 We do not reject H0, i.e., we cannot conclude (at 1% significance level) that the population mean will differ from 10  Because p-value is not quite small, we have weak evidence against H0  reject Ha.

Interpretation of Example B1:

 We reject H0, i.e., we can conclude (at 5% significance level) that the population mean is larger than 0.  Because p-value is quite small, we have very strong evidence against H0  Accept Ha.

Interpretation of Example C1:

 We do not reject H0, i.e., we cannot conclude (at 10% significance level) that the population mean is less than 5.  Because p-value is quite large, we have very weak evidence against H0  reject Ha.

Interpretation of Example A2:

 We __________H0, i.e., we __________ (at 1% significance level) that the population mean will differ from 10  Because p-value is _______________, we have __________evidence against H0  ___________ Ha.

Interpretation of Example B2:

 We __________ H0, i.e., we __________ (at 5% significance level) that the population mean is larger than 0.  Because p-value is _______________, we have __________evidence against H0  __________ Ha. 

Interpretation of Example C2:

 We __________ H0, i.e., we __________ (at 10% significance level) that the population mean is less than 5.  Because p-value is _______________, we have __________evidence against H0  ___________ Ha.

Fill in the empty cells in below table. Use t-table (because sample standard deviation is given)

t-test Example D1 Example D2 Example E1 Example E2 Example F1 Example F2

𝐻 𝜇 = 10 𝜇 = 10 𝜇 ≤ 0 𝜇 ≤ 0 𝜇 ≥ 5 𝜇 ≥ 5 𝐻 𝜇 ≠ 10 𝜇 ≠ 10 𝜇 > 0 𝜇 > 0 𝜇 < 5 𝜇 < 5 𝜇 10 10 0 0 5 5

One-sided or two sided

Two-sided Two-sided One-sided One-sided One-sided One-sided

�̅� 15 18 0.1 0.049 4 4 𝒔 8.84 10 0.13 0.106 1.476 0.935 𝑛 20 20 9 9 15 15

t-statistic 2.53 2.306 -2.624

𝛼 0.01 0.01 0.05 0.05 0.1 0.1

𝛼/2 0.005 0.005 N/A N/A N/A N/A

Critical value (From t table)

2.861 1.86 -1.345

Rejection Region

t>2.86 or

t< -2.86

t>1.86 t<-1.345

Decision

Do not Reject H0 Reject H0 Reject H0

p-value

0.02 0.025 0.01

Degree of evidence against 𝐻

Weak evidence Strong evidence Strong evidence