Question

For a certain drug, based on standards set by the United States Pharmacopeia (USP) - an official public standards-setting authority for all prescription and over-the-counter medicines and other health care products manufactured or sold in the United States, a standard deviation of capsule weights of less than 1.9 mg is acceptable. A sample of 34 capsules was taken and the weights are provided below:

120.5	122.4	118.5	119.7	119.8
119.9	122.1	117.3	119.4	116.2
123.4	119.7	120.2	122.3	119.7
120.7	118.1	122.1	120.1	120.9
120.9	121.8	119.4	123	119.5
116.6	118.8	115.7	122.7	119.6
123.4	120.9	123.2	117.7

(Note: The average and the standard deviation of the data are respectively 120.2 g and 2.06 g.)

At 1% significance level, test the claim that the standard deviation of capsule weights of the drug is greater than 1.9 g.

Procedure: Select an answer One variance χ² Hypothesis Test One proportion Z Hypothesis Test One mean Z Hypothesis Test One mean T Hypothesis Test

Assumptions: (select everything that applies)

• Population standard deviation is unknown
• Sample size is greater than 30
• Simple random sample
• Population standard deviation is known
• Normal population
• The number of positive and negative responses are both greater than 10

Step 1. Hypotheses Set-Up:

H0:H0: Select an answer σ² p μ  =	, where ? p σ μ  is the Select an answer population proportion population standard deviation population mean  and the units are ? mg g lbs kg
Ha:Ha: Select an answer p μ σ²  ? > < ≠	, 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 χ²₀ t₀ f₀  = (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	? yes no

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

Step 6. Interpretation:

At 1% significance level we Select an answer DO NOT DO  have sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

Answer 1

The procedure for the above test is One Variance Chi - Square(²) Hypothesis Test

The Assumptions required for this test are -

• The Population Standard Deviation must be unknown
• The Sample must be obtained using the Simple Random Sample procedure
• The Sample must be taken from a Normal Distribution

Step 1 -

The null hypothesis, H0: ≤ 1.9 g and the alternate hypothesis, Ha: > 1.9 g

and it is a One Sided - Right Tailed test

Step 2 -

The Significance Level, = 1% = 0.01

Step 3 -

The Average / Sample Mean = 120.2 g

Sample Standard Deviation of the data, s' =  2.06 g.

The Size of the Sample, n = 34

The Test Statistic for the above test is = (n - 1)(s'^2) / (^2) which follows Chi - Square Distribution with (n - 1) degrees of freedom under the null hypothesis

Substituting all the values,

The Value of test statistic is 38.792

Step 4 -

The Critical Value, (0.01, 33) = 54.776

P - Value = 0.225

The Test Statistic is not in the rejection region and the P - Value is not less than the Significance Level (0.01)

Generally, if the value of the test statistic exceeds the critical value we reject the null else we fail to reject the null, or if the P - Value is less than the Significance level, we reject the null else we fail to reject the null

Since, P - Value > Significance Level and Value of Test Statistic < Critical Value we fail to reject the null

Conclusion: Do not reject the null hypothesis in favor of the alternative.

Step 5 -

At 1% significance level we DO NOT have sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.