Nine people weight and height.

Weight                                                                Height         

     You will be applying what you have learned in this course by gathering data and running a statistical analysis.

To study the relationship between height and the weight of people you know, you need to collect a sample of nine (9) people using a systematic sampling method.

Where and how are you going to collect your sample?

Collect the sample and record the data.

Construct a confidence interval to estimate the mean height and the mean weight. (CLO 1)

Find the sample mean and the sample standard deviation of the height.

Find the sample mean and the sample standard deviation of the weight.

Construct a confidence interval to estimate the mean height.

Construct a confidence interval to estimate the mean weight.

Test a claim that the mean height of the people is not equal to 64 inches using the p-value method or the traditional method. (CLO 2)

State H0 and H1.

Find the p-value or critical value(s).

Draw a conclusion.

Find the correlation coefficient between the height and the weight.

Construct the equation of the regression line and use it to predict the weight of a person who is 68 inches tall. (CLO 3)

Write one to two paragraphs about what you have learned from this process.

When you read, see, or hear a statistic in the future, what skills will you apply to know whether you can trust the result?

REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults†. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n₁ = 8 children (9 years old) showed that they had an average REM sleep time of x₁ = 2.9 hours per night. From previous studies, it is known that σ₁ = 0.9 hour. Another random sample of n₂ = 8 adults showed that they had an average REM sleep time of x₂ = 2.00 hours per night. Previous studies show that σ₂ = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

(a) State the null and alternate hypotheses.

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.   

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

The Student's t. We assume that both population distributions are approximately normal with known standard deviations.

(c)What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Round your answer to two decimal places.)


(d) Find (or estimate) the P-value. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to the P-value.

(e)

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.    

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

(f) Interpret your conclusion in the context of the application.

Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.     

Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

1.) Consider the data in the following table. In this study, the authors were interested in the use of erythrocyte sodium (ES) concentration as a potential biomarker for the response to lithium treatment in patients with bipolar illness. The ES levels were measured in 10 bipolar patients before and after treatment with lithium. The ES levels before and after treatment with lithium, as well as the after - before differences are given in Table 1. The authors wished to determine if there was a significant increase in ES level following lithium treatment.

            Table 1. Erythrocyte Sodium Concentrations (mmol/l) in Bipolar Patients

Test statistic = __2.19____   d.f. = __9__   p-value = ___ 0.0560___

One sample t test was most appropriate to perform

??? State your conclusion in terms that a layperson would understand. ???

Some sports that involve a significant amount of running, jumping, or hopping put participants at risk for Achilles tendinopathy (AT), an inflammation and thickening of the Achilles tendon. A study looked at the diameter (in mm) of the affected tendons for patients who participated in these types of sports activities. Suppose that the Achilles tendon diameters in the general population have a mean of 5.97 millimeters (mm). When the diameters of the affected tendon were measured for a random sample of 31 patients, the average diameter was 9.70 with a standard deviation of 1.96 mm. Is there sufficient evidence to indicate that the average diameter of the tendon for patients with AT is greater than 5.97 mm? Test at the 5% level of significance.

State the null and alternative hypotheses.

H0: μ = 5.97 versus Ha: μ > 5.97

H0: μ = 5.97 versus Ha: μ < 5.97

H0: μ = 5.97 versus Ha: μ ≠ 5.97

H0: μ ≠ 5.97 versus Ha: μ = 5.97

H0: μ < 5.97 versus Ha: μ > 5.97

Find the test statistic and rejection region. (Round your answers to two decimal places. If the test is one-tailed, enter NONE for the unused region.)

test statistic rejection region

z=

z >

z <

State your conclusion.

H0 is rejected. There is sufficient evidence to indicate that the average diameter of the tendon for patients with AT is greater than 5.97 mm.

H0 is not rejected. There is insufficient evidence to indicate that the average diameter of the tendon for patients with AT is greater than 5.97 mm.

H0 is rejected. There is insufficient evidence to indicate that the average diameter of the tendon for patients with AT is greater than 5.97 mm.

H0 is not rejected. There is sufficient evidence to indicate that the average diameter of the tendon for patients with AT is greater than 5.97 mm.

1 (a-c)

You ask your graduate student to roll a die 5000 times and record the results.

a) Give the expected mean and standard deviation of the outcome.

b) The die roll experiment is repeated (though with a different graduate student – for some reason your previous student went to work with a different advisor). However in this case, the die is weighted so that a 6 shows up 11% of the time, a 1 shows up 25% of the time, and all remaining numbers show up 16% (each). Now what is the expected mean and sd of 5000 rolls?

c) What would happen if we changed the experiment to 10,000 rolls?

Problem 2

An article in the American Industrial Hygiene Association Journal (1976, Vol. 37, pp. 418-422) described a field test for detecting the presence of arsenic in urine samples. The test has been proposed for use among forestry workers because of the increasing use of organic arsenics in that industry. The experiment compared the test as performed by both a trainee and an experienced trainer to an analysis at a remote laboratory. Four subjects were selected for testing and are considered as blocks. The response variable is arsenic content (in ppm) in the subject's urine. The data are as follows. Use Excel to do an ANOVA to determine if there is any difference between the Trainee, Trainer and Lab in measuring arsenic.

Part A.) Find the data for Problem 2a in the Ch. 13 homework problem data set file posted to Bb in the Ch. 13 folder. Use Excel to perform the ANOVA. In looking at the data, what is the definition of an entry that reads ".05"?

multiple choice:

A. It is the fraction of samples that were inspected by a trainer or trainee
B. It is the percent accuracy of a trainer or a trainee
C. It is the measured arsenic content in a subject's urine
D.It is the measured arsenic content in the urine of a trainee or trainer

E.None of the above

Part B.) What is the p-value for the hypothesis test associated with Problem 2?

Part C.) What will be the conclusion of the hypothesis test associated with Problem 2?

A. There is no difference in mean arsenic levels in the urine of the four subjects

B. There is no difference in mean arsenic levels in the urine of the trainer, trainee and lab

C. One of the subjects has a higher concentration of arsenic in his/her urine

D. The trainer, trainee and lab, when measuring arsenic in urine, are all yielding the same mean measured arsenic level.

E. none of the above

Using R Studio.

The dataset weightloss.txt presents data for the weight loss of a compound for different amounts of time the compound was exposed to the air. Additional information was also available on the humidity of the environment during exposure. The relative humidity has been coded as A= 20%, B= 30% and C=40% humidity and the dummy variables x2 and x3 have been formed to code humidity accordingly.

a)Determine and overall simple LSRL model for predicting weight based off of time and humidity (variables x2 and x3). Is this model a “good” model? Explain.

b)Create a scatterplot (change colors) to illustrate that there appears to exist an interaction effect between time and level of humidity.

c)Create a regression model that includes the appropriate interaction terms. Does this model appear to be a “better” model than part (a)? Explain.

d)Using the regression coefficients found in part (c), find the “best” fitting linear model for each of the three humidity levels

e).Plot each of the three lines found in part (d) onto the original scatterplot along with the line from part (a).

weightloss.txt

> weight<-c(7.3,6.5,5.1,4,4,5.2,6.6,6.6,2,4,5.7,6.5)
> time<-c(4,5,6,7,4,5,6,7,4,5,6,7)
> x2<-c(1,1,1,1,0,0,0,0,0,0,0,0)
> x3<-c(0,0,0,0,1,1,1,1,0,0,0,0)

> humidity<-c('A','A','A','A','B','B','B','B','C','C','C','C')

5. A cruise company would like to estimate the average beer consumption to plan its beer inventory levels on future​ seven-day cruises.​ (The ship certainly​ doesn't want to run out of beer in the middle of the​ ocean!) The average beer consumption over 12 randomly selected​ seven-day cruises was 81,845 bottles with a sample standard deviation of 4,528. complete parts a and b below.

a. Construct a 90​% confidence interval to estimate the average beer consumption per cruise.

lower limit of ___ bottles to an upper limit of ___ bottles

b. . What assumptions need to be made about this​ population?

6. A national air traffic control system handled an average of 47,574 flights during 28 randomly selected days in a recent year. The standard deviation for this sample is 6,421 flights per day. Complete parts a through c below.

a. Construct a 99​% confidence interval to estimate the average number of flights per day handled by the system.

The 99​% confidence interval to estimate the average number of flights per day handled by the system is from a lower limit of ___ to an upper limit of ___

b. Suppose an airline company claimed that the national air traffic control system handles an average of​ 50,000 flights per day. Do the results from this sample validate the airline​ company's claim?

c. What assumptions need to be made about this​ population?

Describe the similarities and differences between

A.    A prospective cohort study and a retrospective cohort study

B.    A prospective cohort study and a case control study

C    A case control study and retrospective cohort study

D.    Is a case control study always retrospective? Why or why not?

A student group claims that first-year students at a university should study 2.5 hours (150 minutes) per night during the school week. A skeptic suspects that they study less than that on the average. A survey of 51 randomly selected students finds that on average students study 138 minutes per night with a standard deviation of 32 minutes. What conclusion can be made from this data? Select one:

A) The p-value is greater than .05, therefore we do not have enough evidence to conclude that students study less than 150 minutes per night.

B) The p-value is less than .05, therefore we conclude that students study greater than 150 minutes per night.

C) The p-value is less than .05, therefore we conclude that students study less than 150 minutes per night.

D) We do not have enough information to make a conclusion about this study. The p-value is less than .05, therefore we do not have enough evidence to conclude that students study less than 150 minutes per night.

Question 25)

A particular fruit's weights are normally distributed, with a mean of 771 grams and a standard deviation of 28 grams.

If you pick 5 fruit at random, what is the probability that their mean weight will be between 750 grams and 779 grams

A study is done to test the claim that Company A retains its workers longer than Company B. Company A samples 16 workers, and their average time with the company is 5.2 years with a standard deviation of 1.1. Company B samples 21 workers, and their average time with the company is 4.6 years with a standard deviation of 0.9. The populations are normally distributed.
write the hypothesis in symbolic form.
determine the value of the test statistic.
find the critical value OR the p-value.
determine if you should reject the null hypothesis or fail to reject the null hypothesis.
write a conclusion addressing the original claim.
All work must be shown in order to receive any credit. 

Assume that 60% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places:

a. There are some lefties (≥ 1) among the 5 people.

b. There are exactly 3 lefties in the group.

c. There are at least 4 lefties in the group.

d. There are no more than 2 lefties in the group.

e. How many lefties do you expect?

f. With what standard deviation?

For one binomial experiment,

n₁ = 75

binomial trials produced

r₁ = 30

successes. For a second independent binomial experiment,

n₂ = 100

binomial trials produced

r₂ = 50

successes. At the 5% level of significance, test the claim that the probabilities of success for the two binomial experiments differ. (a) Compute the pooled probability of success for the two experiments. (Round your answer to three decimal places.)


(b) Check Requirements: What distribution does the sample test statistic follow? Explain.

The Student's t. The number of trials is sufficiently large. The standard normal. We assume the population distributions are approximately normal.     The Student's t. We assume the population distributions are approximately normal. The standard normal. The number of trials is sufficiently large.

(c) State the hypotheses.

H₀: p₁ = p₂; H₁: p₁ < p₂H₀: p₁ = p₂; H₁: p₁ ≠ p₂     H₀: p₁ < p₂; H₁: p₁ = p₂H₀: p₁ = p₂; H₁: p₁ > p₂

(d) Compute p̂₁ - p̂₂.
p̂₁ - p̂₂ =

Compute the corresponding sample distribution value. (Test the difference p₁ − p₂. Do not use rounded values. Round your final answer to two decimal places.)


(e) Find the P-value of the sample test statistic. (Round your answer to four decimal places.)


(f) Conclude the test.

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.     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 reject the null hypothesis and conclude the data are statistically significant.

(g) Interpret the results.

Reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ. Reject the null hypothesis, there is insufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.     Fail to reject the null hypothesis, there is insufficient evidence that the probabilities of success for the two binomial experiments differ. Fail to reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.