Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 50 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.40 ml/kg for the distribution of blood plasma.
(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)
| lower limit | |
| upper limit | |
| margin of error |
(b) What conditions are necessary for your calculations? (Select
all that apply.)
n is large
σ is known
the distribution of weights is normal
σ is unknown
the distribution of weights is uniform
(c) Interpret your results in the context of this problem.
1% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.
99% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.
The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.
The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.
(d) Find the sample size necessary for a 99% confidence level with
maximal margin of error E = 2.60 for the mean plasma
volume in male firefighters. (Round up to the nearest whole
number.)
male firefighters
In: Math
There are many controversies in the real life about the hypothesis testing. What do you think are the potential reasons for that? Can we find and share any example of misleading results of the hypothesis testing?
In: Math
Adam, Brian, Chris and Donald are taking goal shots in soccer. They take shots in the order, Adam, Brian, Chris, Donald, then repeat. By going in this order they have an equal chance of making a goal first. Donald makes a goal 50% of the time. What would be the probability of Adam winning if he went last instead of first?
In: Math
When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.
Method 1: Use the Student's t distribution with
d.f. = n − 1.
This is the method used in the text. It is widely employed in
statistical studies. Also, most statistical software packages use
this method.
Method 2: When n ≥ 30, use the sample standard
deviation s as an estimate for σ, and then use
the standard normal distribution.
This method is based on the fact that for large samples, s
is a fairly good approximation for σ. Also, for large
n, the critical values for the Student's t
distribution approach those of the standard normal
distribution.
Consider a random sample of size n = 31, with sample mean x = 44.7 and sample standard deviation s = 5.4.
(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.
| 90% | 95% | 99% | |
| lower limit | |||
| upper limit |
(b) Compute 90%, 95%, and 99% confidence intervals for μ
using Method 2 with the standard normal distribution. Use
s as an estimate for σ. Round endpoints to two
digits after the decimal.
| 90% | 95% | 99% | |
| lower limit | |||
| upper limit |
(c) Compare intervals for the two methods. Would you say that
confidence intervals using a Student's t distribution are
more conservative in the sense that they tend to be longer than
intervals based on the standard normal distribution?
Yes. The respective intervals based on the t distribution are longer.
No. The respective intervals based on the t distribution are longer.
Yes. The respective intervals based on the t distribution are shorter.
No. The respective intervals based on the t distribution are shorter.
(d) Now consider a sample size of 71. Compute 90%, 95%, and 99%
confidence intervals for μ using Method 1 with a Student's
t distribution. Round endpoints to two digits after the
decimal.
| 90% | 95% | 99% | |
| lower limit | |||
| upper limit |
(e) Compute 90%, 95%, and 99% confidence intervals for μ
using Method 2 with the standard normal distribution. Use
s as an estimate for σ. Round endpoints to two
digits after the decimal.
| 90% | 95% | 99% | |
| lower limit | |||
| upper limit |
(f) Compare intervals for the two methods. Would you say that
confidence intervals using a Student's t distribution are
more conservative in the sense that they tend to be longer than
intervals based on the standard normal distribution?
Yes. The respective intervals based on the t distribution are shorter.
No. The respective intervals based on the t distribution are longer.
No. The respective intervals based on the t distribution are shorter.
Yes. The respective intervals based on the t distribution are longer.
With increased sample size, do the two methods give respective
confidence intervals that are more similar?
As the sample size increases, the difference between the two methods remains constant.
As the sample size increases, the difference between the two methods becomes greater.
As the sample size increases, the difference between the two methods is less pronounced.
In: Math
State H0 and H1. Answer the question. Use megastat and submit software output.
See Worksheet 4. John Isaac Inc., a designer and installer of industrial signs, employs 60 people. The company recorded the type of the most recent visit to a doctor by each employee. A national assessment conducted in 2014 found that 53% of all physician visits were to primary care physicians, 19% to medical specialists, 17% to surgical specialists, and 11% to emergency departments. Test at the .01 significance level if Isaac employees differ significantly from the national survey distribution.
| National | Problem 6 | ||
| Proportion | Visit Type | Observed | Expected |
| 0.53 | Primary care | 29 | |
| 0.19 | Medical specialist | 11 | |
| 0.17 | Surgical specialist | 16 | |
| 0.11 | Emergency | 4 | |
| 60 | |||
In: Math
The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages.
| 1.6 | 2.4 | 1.2 | 6.6 | 2.3 | 0.0 | 1.8 | 2.5 | 6.5 | 1.8 |
| 2.7 | 2.0 | 1.9 | 1.3 | 2.7 | 1.7 | 1.3 | 2.1 | 2.8 | 1.4 |
| 3.8 | 2.1 | 3.4 | 1.3 | 1.5 | 2.9 | 2.6 | 0.0 | 4.1 | 2.9 |
| 1.9 | 2.4 | 0.0 | 1.8 | 3.1 | 3.8 | 3.2 | 1.6 | 4.2 | 0.0 |
| 1.2 | 1.8 | 2.4 |
(a) Use a calculator with mean and standard deviation keys to find x and s. (Round your answers to two decimal places.)
| x = | % |
| s = | % |
(b) Compute a 90% confidence interval for the population mean
μ of home run percentages for all professional baseball
players. Hint: If you use the Student's t
distribution table, be sure to use the closest d.f. that
is smaller. (Round your answers to two decimal
places.)
| lower limit | % |
| upper limit | % |
(c) Compute a 99% confidence interval for the population mean
μ of home run percentages for all professional baseball
players. (Round your answers to two decimal places.)
| lower limit | % |
| upper limit | % |
(d) The home run percentages for three professional players are
below.
| Player A, 2.5 | Player B, 2.3 | Player C, 3.8 |
Examine your confidence intervals and describe how the home run percentages for these players compare to the population average.
We can say Player A falls close to the average, Player B is above average, and Player C is below average.
We can say Player A falls close to the average, Player B is below average, and Player C is above average.
We can say Player A and Player B fall close to the average, while Player C is above average.
We can say Player A and Player B fall close to the average, while Player C is below average.
(e) In previous problems, we assumed the x distribution
was normal or approximately normal. Do we need to make such an
assumption in this problem? Why or why not? Hint: Use the
central limit theorem.
Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.
Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.
No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.
No. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.
In: Math
| Price | Bedroom | Bathroom | Cars | SQ FT |
| 298,000 | 3 | 2.5 | 0 | 1,566 |
| 319,900 | 3 | 2.5 | 0 | 2,000 |
| 354,000 | 3 | 2 | 2 | 0 |
| 374,900 | 4 | 2.5 | 0 | 2,816 |
| 385,000 | 4 | 2 | 0 | 0 |
| 389,000 | 3 | 2.5 | 0 | 2,248 |
| 399,000 | 4 | 3 | 0 | 2,215 |
| 415,000 | 3 | 2.5 | 0 | 3,188 |
| 444,900 | 3 | 2 | 0 | 2,530 |
| 450,000 | 3 | 2 | 0 | 1,967 |
| 465,000 | 4 | 3 | 0 | 2,564 |
| 340,000 | 4 | 2.5 | 0 | 2,293 |
| 275,000 | 3 | 2.5 | 2 | 1,353 |
| 425,000 | 3 | 2 | 0 | 1,834 |
| 250,000 | 3 | 2.5 | 0 | 5,837 |
| 450,000 | 3 | 2.5 | 0 | 9,060 |
| 390,000 | 3 | 3.5 | 0 | 1,002 |
| 269,000 | 3 | 2.5 | 0 | 1,680 |
| 425,000 | 3 | 2.5 | 2 | 4,356 |
| 425,000 | 2 | 2.5 | 2 | 2,993 |
| 425,000 | 3 | 3 | 0 | 4,356 |
| 429,900 | 5 | 3.5 | 1 | 2,154 |
| 400,000 | 3 | 2.5 | 2 | 1,846 |
| 399,900 | 3 | 2 | 1 | 2,018 |
| 388,990 | 4 | 4 | 0 | 2,295 |
In: Math
Listed below are the log body weights and log brain weights of the primates species in the data set ”mammals”. Find the equation of the least squares line with y = log brain weight and x = log body weight. Do it by hand, by constructing a table like the one in Example 9.1. Then do it with your calculator as efficiently as possible. Finally, use the lm function in R to do it by creating a linear model object ”primates.lm”. The model formula is ”log(brain)∼log(body)”. You can select the primates and put them in a new data frame by first listing the primate species names:
> primatenames=c(”Owl monkey”, ”Patas monkey”, ”Gorilla”, etc.)
and then
> primates=mammals[primatenames, ]
Your ”data” argument in calling lm would be ”data=primates”, as in
> primates.lm=lm(log(brain)∼log(body),data=primates)
Alternatively, you can just use ”mammals[primatenames, ]” as the data argument in lm, that is,
> primates.lm=lm(log(brain)∼log(body), data=mammals[primatenames,])
log body log brain
Owl monkey -0.7339692 2.740840
Patas monkey 2.3025851 4.744932
Gorilla 5.3327188 6.006353
Human 4.1271344 7.185387
Rhesus monkey 1.9169226 5.187386
Chimpanzee 3.9543159 6.086775
Baboon 2.3561259 5.190175
Verbet 1.4327007 4.060443
Galago -1.6094379 1.609438
Slow loris 0.3364722 2.525729
In: Math
Let x be a random variable that represents red blood cell (RBC) count in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal. For the population of healthy female adults, the mean of the x distribution is about 4.8 (based on information from Diagnostic Tests with Nursing Implications, Springhouse Corporation). Suppose that a female patient has taken six laboratory blood tests over the past several months and that the RBC count data sent to the patient’s doctor are 4.9, 4.2, 4.5, 4.1, 4.4, 4.3
(a) Use a calculator to verify that ?̅= 4.40 and ? = 0.28.
(b) Do the given data indicate that the population mean RBC count for this patient is lower than 4.8? Use ? = .05.
(c) Obtain a 90% confidence interval for the mean RBC count.
(d) Can you come to the same conclusion you made in (b) using the interval approach in (c).
In: Math
What is the impact of widening the margin of error?
Question 9 options:
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The confidence level increases. |
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The confidence level decreases. |
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The confidence level stays the same. |
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It is unclear what the confidence level does. You dropped out of DePaul to pursue your passion as an amateur entomologist. Congrats,...I guess. You collect 25 cockroaches and measure the width of their thorax. The mean is 1.7 cm and you know the standard deviation of the population's thorax width is 0.25. Calculate the 95% confidence interval. Question 10 options:
|
In: Math
The management at an environmental consulting firm claims the mean weekly salary is $275 with a standard deviation of $34.10.
If we want to address the question “If management’s claims are true, what is the probability that an individual worker would make an average weekly salary less than $264.50?”, write the probability statement.
If we want to address the question “If management’s claims are true, what is the probability that an individual worker would make an average weekly salary less than $264.50?”, what is the probability? Be sure to show how you calculated your probability.
If we want to address the question “If management’s claims are true, what is the probability that a group of 27 workers would make an average weekly salary less than $264.50?”, write the probability statement.
If we want to address the question “If management’s claims are true, what is the probability that a group of 27 workers would make an average weekly salary less than $264.50?”, what is the probability? Be sure to show how you calculated your probability.
In: Math
Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. Let x be a random variable that represents the number of milligrams of porphyrin per deciliter of blood. In healthy circles, x is approximately normally distributed with mean μ = 44 and standard deviation σ = 14. Find the following probabilities. (Round your answers to four decimal places.)
(a) x is less than 60
(b) x is greater than 16
(c) x is between 16 and 60
(d) x is more than 60 (This may indicate an infection,
anemia, or another type of illness.)
In: Math
Let n1equals100, Upper X 1equals50, n2equals100, and Upper X 2equals30. Complete parts (a) and (b) below. a. At the 0.01 level of significance, is there evidence of a significant difference between the two population proportions?
a) Calculate the test statistic, Upper Z Subscript STAT, based on the difference p1minusp2. The test statistic, Upper Z Subscript STAT.
b) While either a standardized normal distribution table or technology may be used to calculate the p-value, for this exercise, use technology. Identify the value of the p-value from your technology output.
c. Construct a 95% confidence interval estimate of the difference between the two population proportions.
In: Math
In: Math
Q6. You are the assistant director of political research for the NBC television network, and two candidates Jeffrey Temple and Rotenberg Marvel, are running for president of the United States. You need to furnish a prediction of the percentage of the vote going to Marin, assuming the election was held today, for tomorrow’s evening newscast. You want to be 93% confident in your prediction and desire a total precision of ±4 percentage points.
a) Assume that you have no reliable information concerning the percentage of the population that prefers Marvel. What sample size will you use for the project? Hint: in this case, you should be conservative assuming the maximum degree of heterogeneity in the population so as to have a sample that is larger than it is necessary.
b) Assume that a similar poll, taken thirty days ago, revealed that 43 percent of the respondents would vote for Marvel. Taking this information into account, what sample size will you use for the project?
In: Math