Question

1) A pharmaceutical company is about to launch a new manufacturing process in addition to the existing one. The quality control manager believes that the new method results in a different variation in the weights of the capsules. To verify the claim, the samples from each production line were obtained and the results are below (in mg):

Production Line 1:

101.4	102.5	100.2	98.3	100.3
101.2	98.8	98.2	99	99.6
101.1	101.9	102.3	100.3	98.4
99.3	102.2

(Note: The average and the standard deviation of the data are respectively 100.3 mg and 1.48 mg.)

Production Line 2:

97.6	99.9	98.6	97.9	97.2
97.7	98	98.6	97.9	99.5
99.6	99	99.5	99.2

(Note: The average and the standard deviation of the data are respectively 98.6 mg and 0.87 mg.)

Use a 5% significance level to test the claim that the standard deviation of the capsule weights in the production line 1 is greater than the standard deviation of the capsule weights in the production line 2.

Procedure: Select an answer Two means Z Hypothesis Test Two paired means Z Hypothesis Test Two proportions Z Hypothesis Test Two means T (pooled) Hypothesis Test Two means T (non-pooled) Hypothesis Test Two variances F Hypothesis Test

Assumptions: (select everything that applies)

• Population standard deviation are unknown but assumed equal
• Population standard deviation are unknown
• Paired samples
• Sample sizes are both greater than 30
• Normal populations
• The number of positive and negative responses are both greater than 10 for both samples
• Independent samples
• Population standard deviations are known
• Simple random samples

Step 1. Hypotheses Set-Up:

H0:H0: Select an answer p₁-p₂ μ μ₁-μ₂ σ₁²/σ₂²  =	, where Select an answer σ's are μ's are μ=μ₁-μ₂ is p's are  the Select an answer population means difference between population means population standard deviations population proportions  and the units are ? kg mg oz lbs
Ha:Ha: Select an answer σ₁²/σ₂² μ p₁-p₂ μ₁-μ₂  ? > ≠ <	, and the test is Select an answer Two-Tail Right-Tail Left-Tail

Step 2. The significance level α=α= %

Step 3. Compute the value of the test statistic: Select an answer z₀ t₀ f₀ χ²₀  = (Round the answer to 3 decimal places)

Step 4. Testing Procedure: (Round the answers to 3 decimal places)

CVA	PVA
Provide the critical value(s) for the Rejection Region:	Compute the P-value of the test statistic:
left CV is  and right CV is	P-value is

Step 5. Decision:

CVA	PVA
Is the test statistic in the rejection region?	Is the P-value less than the significance level?
? yes no	? no yes

Conclusion: Select an answer Do not reject the null hypothesis in favor of the alternative. Reject the null hypothesis in favor of the alternative.

Step 6. Interpretation:

At 5% significance level we Select an answer DO NOT DO  have sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

1. What is the conclusion?
   • There is not sufficient evidence to support the claim that 20% of people support the death penalty has decreased.
   • There is sufficient evidence to support the claim that 20% of people support the death penalty has decreased.
   
   

Question 2

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The proportion of people that own cats is 50%. A veterinarian believes that this proportion is larger than 50% and surveys 400 people. Test the veterinarian's claim at the α=0.01α=0.01 significance level.

Preliminary:

1. Is it safe to assume that n≤0.05n≤0.05 of all subjects in the population?
   • Yes
   • No
   
   
2. Verify np(1−p)≥10.np(1-p)≥10. Round your answer to one decimal place.
   
   np(1−p)=np(1-p)=

Test the claim:

1. The null and alternative hypotheses are
   • H0:μ=0.5H0:μ=0.5
     Ha:μ≠0.5Ha:μ≠0.5
   • H0:μ≥0.5H0:μ≥0.5
     Ha:μ<0.5Ha:μ<0.5
   • H0:p=0.5H0:p=0.5
     Ha:p<0.5Ha:p<0.5
   • H0:μ≤0.5H0:μ≤0.5
     Ha:μ>0.5Ha:μ>0.5
   • H0:p≤0.5H0:p≤0.5
     Ha:p>0.5Ha:p>0.5
   • H0:p=0.5H0:p=0.5
     Ha:p≠0.5Ha:p≠0.5
   
   
2. The test is
   Select an answer two-tailed right-tailed left-tailed
   
3. Based on the sample of 400 people, 57% owned cats. Calculate the test statistic. Round to two decimal places.
   z=
   
4. Calculate the p -value. Round to four decimal places.
   
   
5. Make a decision.
   • Reject the null hypothesis
   • Fail to reject the null hypothesis
   
   
6. Make a conclusion.
   • There is sufficient evidence that the proportion of people who own cats is larger.
   • There is not sufficient evidence that the proportion of people who own cats is larger.

Please also provide excel formulas that are used to calculate the  Left and Right CVA, Z- score and P value

Answer 1

1. Option: Two variances F Hypothesis Test

Option: i. Normal populations; ii. Independent samples

Step 1. Hypotheses Set-Up:

(population standard deviations and unit mg).

Step 2. The significance level α=5%

Step 3. f₀=2.1956/0.7644=2.872

Step 4:

Step 5. Decision: Yes, the test statistic is in the rejection region.

Yes, P-value is less than the significance level.

Conclusion: Reject the null hypothesis in favor of the alternative.

Step 6. Interpretation:

At 5% significance level we Select an answer DO  have sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

Conclusion: There is sufficient evidence to support the claim that the standard deviation of the capsule weights in the production line 1 is greater than the standard deviation of the capsule weights in the production line 2.

2. Yes, it is safe to assume that n≤0.05n≤0.05 of all subjects in the population.

n=400, p=0.5, np=200, n(1-p)=200

np(1-p)=400*0.5*0.5=100.

Test the claim: The null and alternative hypotheses are H₀:p≤0.5, H_a:p>0.5

The test is right-tailed.

Since P-value<0.01, Reject the null hypothesis.

There is sufficient evidence that the proportion of people who own cats is larger.