In Problems 8-12, please use the following steps (i)-(v) for all hypothesis tests:
(i) What is the level of significance? State the null and alternate hypotheses.
(ii) Check Requirements What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic?
(iii) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.
(iv) Based on your answers in parts (i)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a?
(v) Interpret your conclusion in the context of the application.
Note: For degrees of freedom d.f. not in the Student’s t table, use the closet d.f. that is smaller. In some situation, this choice of d.f. may increase the P-value a small amount and thereby produce a slightly more "conservative” answer.
Testing and Estimating µ with σ Unknown Carboxyhemoglobin is formed when hemoglobin is exposed to carbon monoxide. Heavy smokers tend to have a high percentage of carboxyhemoglobin in their blood (Reference: A Manual of Laboratory and Diagnostic Tests by F. Fischbach). Let x be a random variable representing percentage of carboxyhemoglobin in the blood. For a person who is a regular heavy smoker, x has a distribution that is approximately normal. A random sample of n = 12 blood tests given to a heavy smoker gave the following results (percentage of carboxyhemoglobin in the blood):
9.1 | 9.5 | 10.2 | 9.8 | 11.3 | 12.2 |
11.6 | 10.3 | 8.9 | 9.7 | 13.4 | 9.9 |
(a) Use a calculator to verify that
(b) A long-term population
(c) Use the given data to find a 99% confidence interval for µ for this patient.
(a)
Whether the sample mean
Answer to Problem 9CRP
Solution:
Yes, the sample mean
Explanation of Solution
To calculate the required statistics using the Minitab, follow the below instructions:
Step 1: Go to the Minitab software.
Step 2: Go to Stat >Basic statistics > Display Descriptive Statistics.
Step 3: Select ‘Percentages’ in variables.
Step 4: Click on OK.
The obtained statistics is:
Descriptive Statistics: Percentages
Statistics
Variable | N | N* | Mean | SE Mean | StDev | Minimum | Q1 | Median | Q3 | Maximum |
Percentages | 12 | 0 | 10.492 | 0.392 | 1.358 | 8.900 | 9.550 | 10.050 | 11.525 | 13.400 |
From the Minitab output, the sample mean and sample standard deviation are approximately equals to
(b)
(i)
The level of significance, null and alternative hypothesis.
Answer to Problem 9CRP
Solution:
The level of significance is
Explanation of Solution
The level of significance is defined as the probability of rejecting the null hypothesis when it is true, it is denoted by
Null hypothesis
Alternative hypothesis
(ii)
To find:
The sampling distribution that should be used and compute the value of the sample test statistic.
Answer to Problem 9CRP
Solution:
The student's t distribution should be used. The sample test statisticis 1.25.
Explanation of Solution
Calculation:
We assume that x distribution is mound shape and symmetrical, because
Using
The sample test statistic t is
(iii)
To find:
The P-value of the test statistic and sketch the sampling distribution showing the area corresponding to the P-value.
Answer to Problem 9CRP
Solution:
The P-value of the test statistic is 0.1186.
Explanation of Solution
Calculation:
We have t = 1.25
Using Table 4 from the Appendix to find the specified area:
Graph:
To draw the required graphs using the Minitab, follow the below instructions:
Step 1: Go to the Minitab software.
Step 2: Go to Graph > Probability distribution plot > View probability.
Step 3: Select ‘t’ and enter d.f = 11.
Step 4: Click on the Shaded area > X value.
Step 5: Enter X-value as 1.25 and select ‘Right Tail’.
Step 6: Click on OK.
The obtained distribution graph is:
(iv)
Whether we reject or fail to reject the null hypothesisand whether the data is statistically significant for a level of significance of 0.05.
Answer to Problem 9CRP
Solution:
The P-value
Explanation of Solution
The P-value of 0.1186 is greater than the level of significance (α) of 0.05. Therefore we failed to reject the null hypothesis
(v)
The interpretation for the conclusion.
Answer to Problem 9CRP
Solution:
There is not enough evidence to conclude that sample mean of 12 smokers' Carboxyhoemoglobin levels is significantly large enough to sound a clinical alert.
Explanation of Solution
The P-value is greater than the level of significance (
(b)
To find:
A 99% confidence interval for the mean for this patient.
Answer to Problem 9CRP
Solution:
The 99% confidence interval for
Explanation of Solution
Calculation:
We have to find 99% confidence interval
Then, the 99% confidence interval is
The 99% confidence interval for
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Chapter 9 Solutions
Understanding Basic Statistics
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