Carboxyhemoglobin is formed when hemoglobin is exposed to carbon monoxide. Heavy smokers tend to have a high percentage of carboxyhemoglobin in their blood.† 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 (percent carboxyhemoglobin in the blood). Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and thereby produce a slightly more "conservative" answer. 9.3 9.7 10.5 9.8 11.3 12.2 11.6 10.3 8.9 9.7 13.4 9.9 (a) Use your calculator to calculate x and s. (Round your answers to two decimal places.) x = s = (b) A long-term population mean μ = 10% is considered a health risk. However, a long-term population mean above 10% is considered a clinical alert that the person may be asymptomatic. Do the data indicate that the population mean percentage is higher than 10% for this patient? Use α = 0.05. (i) What is the level of significance? State the null and alternate hypotheses. H0: μ > 10%; H1: μ = 10%H0: μ = 10%; H1: μ < 10% H0: μ = 10%; H1: μ ≠ 10%H0: μ = 10%; H1: μ > 10% (ii) What sampling distribution will you use? What assumptions are you making? The Student's t, since we assume that x has a normal distribution with known σ.The standard normal, since we assume that x has a normal distribution with unknown σ. The Student's t, since we assume that x has a normal distribution with unknown σ.The standard normal, since we assume that x has a normal distribution with known σ. What is the value of the sample test statistic? (Round your answer to three decimal places.) (iii) Find (or estimate) the P-value. P-value > 0.2500.125 < P-value < 0.250 0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005 Sketch the sampling distribution and show the area corresponding to the P-value.(iv) Based on your answers in parts (i) to (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (v) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the patient is asymptotic.There is insufficient evidence at the 0.05 level to conclude that the patient is asymptotic. (c) Use the given data to find a 99% confidence interval for μ for this patient. (Round your answers to two decimal places.) lower limit % upper limit %
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Carboxyhemoglobin is formed when hemoglobin is exposed to carbon monoxide. Heavy smokers tend to have a high percentage of carboxyhemoglobin in their blood.† 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
tests given to a heavy smoker gave the following results (percent carboxyhemoglobin in the blood). Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and thereby produce a slightly more "conservative" answer.
9.3 | 9.7 | 10.5 | 9.8 | 11.3 | 12.2 |
11.6 | 10.3 | 8.9 | 9.7 | 13.4 | 9.9 |
x | = | |
s | = |
(b) A long-term population mean
State the null and alternate hypotheses.
(ii) What sampling distribution will you use? What assumptions are you making?
What is the value of the sample test statistic? (Round your answer to three decimal places.)
(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) to (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?
(v) Interpret your conclusion in the context of the application.
(c) Use the given data to find a 99% confidence interval for μ for this patient. (Round your answers to two decimal places.)
lower limit | % |
upper limit | % |
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