Analysis of variance (ANOVA)

An investigator compares the difference in the amount of 20% sucrose solution removed by pollinators between unadulterated solution and solutions with 50, 100, 150, and 200 parts per million of caffeine. The results are expressed as differences in grams for the following caffeine concentrations:

50 ppm caffeine: -0.4 0.34. 0.19, 0.05, -0.14

100 ppm caffeine: 0.01, -0.39, -0.08, -0.09, -0.31

150 ppm caffeine: 0.65, 0.53, 0.39, -0.15, 0.46

200 ppm caffeine: 0.24, 0.44, 0.13, 1.03, 0.05

Does the mean amount of nectar taken depend on the concentration of caffeine in the nectar?

Carry out an analysis of variance (ANOVA) to find out.

a. State the null and alternative hypotheses appropriate for this question.

b. Calculate the following summary statistics for each group: n_i, Y-bar_i, and s_i.

c. Set up an ANOVA table to keep track of your results.

d. What is the mean square error MS_error?

e. How many degrees of freedom are associated with error? How many with groups?

f. Calculate the estimate of the grant mean.

g. Calculate the group sum of squares.

h. Calculate the group degrees of freedom and the group mean square.

i. What is F for this example?

j. Use a statistical table or Excel to find the P-value for this test.

Consider the following results for independent samples taken from two populations.

a. What is the point estimate of the difference between the two population proportions (to 2 decimals)?  

b. Develop a 90% confidence interval for the difference between the two population proportions (to 4 decimals).  
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c. Develop a 95% confidence interval for the difference between the two population proportions (to 4 decimals).  
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Approximate ?(68.5 < ?̅ < 71.5) when ?̅ is the mean of a random sample of size 12 from a distribution with ? = 70 and variance of 75.

Today, the waves are crashing onto the beach every 5.8 seconds. The times from when a person arrives at the shoreline until a crashing wave is observed follows a Uniform distribution from 0 to 5.8 seconds. Round to 4 decimal places where possible.

1)The probability that wave will crash onto the beach exactly 1.5 seconds after the person arrives is P(x = 1.5) = The probability that it will take longer than 4.06 seconds for the wave to crash onto the beach after the person arrives is P(x > 4.06) =

2)Suppose that the person has already been standing at the shoreline for 0.7 seconds without a wave crashing in. Find the probability that it will take between 2.9 and 5.4 seconds for the wave to crash onto the shoreline.  ________

3)62% of the time a person will wait at least how long before the wave crashes in? ___________seconds.

4)Find the minimum for the lower quartile. _____________ seconds.

The authors of a paper studied a random sample of 351 Twitter users. For each Twitter user in the sample, the tweets sent during a particular time period were analyzed and the Twitter user was classified into one of the following categories based on the type of messages they usually sent.

The accompanying table gives the observed counts for the five categories (approximate values read from a graph in the paper).

Carry out a hypothesis test to determine if there is convincing evidence that the proportions of Twitter users falling into each of the five categories are not all the same. Use a significance level of

α = 0.05.

(Hint: See Example 14.3.)

Let p₁, p₂, p₃, p₄, and p₅ be the proportions of Twitter users falling into the five categories.

State the appropriate null and alternative hypotheses.

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 0.5
H_a: H₀ is not true. H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 351
H_a: H₀ is not true.     H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 0.2
H_a: H₀ is not true. H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 70
H_a: H₀ is not true. H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 0.05
H_a: H₀ is not true.

Find the test statistic and P-value. (Use technology. Round your test statistic to three decimal places and your P-value to four decimal places.)

X² = P-value =

State the conclusion in the problem context.

Do not reject H₀. There is not convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same. Reject H₀. There is not convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same.     Reject H₀. There is convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same. Do not reject H₀. There is convincing evidence to conclude that the proportions of Twitter users falling into the five categories are not all the same.

Mr Ahuja have always been interested in whether or not where a student sits is related to the students overall grades in school. Below is a table that divides students into 3 seating areas: Front, Middle, and Back with their given GPAS.

Front Middle Back

3.062 2.859 2.583

3.894 2.639 2.653

2.966 3.634 3.09

3.575   3.564 3.06

4   2.115 2.463

2.69 3.08 2.598

3.523 2.937 2.879

3.332 3.091 2.926

3.885 2.655 3.221

3.559 2.526 2.646

Calculate a One-Way ANOVA table (using EXCEL) for the data above. Complete the following: At α = .05, test to see if there is a significant difference among the average GPA of all the students based on three areas of seating. Use both the critical and p-value approaches. Include hypotheses, critical values, results, and conclusions in the language of the problem.

Q1. Does an average box of cereal contain 368 grams of cereal? A random sample of 36 boxes had a mean of 372.5 and a standard deviation of 12 grams. Test at the .05 level of significance.

A box contains three white balls, two black balls, and one red ball. Three balls are drawn at random without replacement. Let Y1 be the number of white balls drawn, and Y2 the number of black balls drawn. Find the distribution of Z = Y1 × Y2

1.  An investigator wishes to compare the average time to relief of headache pain under three

distinct medications, call them Drugs A, B, and C. Fifteen patients who suffer from chronic

headaches are randomly selected for the investigation, and five subjects are randomly assigned

to each treatment. The following data reflect times to relief (in minutes) after taking the

assigned drug.

Test if there is a significant difference in the mean times among three treatments. Use α =

0.05. (apply ANOVA)

a) State the null and alternative hypothesis.

Null: all means are equal

Alternative: Atleast one mean is different

b) State your conclusion about the hypothesis based on the test statistic and critical value

Reject the hypothesis because there is a huge difference between the 3 treatments and their means

c) Perform multiple comparison test to show whether there are significant differences between the drugs

For the following data​ (a) display the data in a scatter​ plot, (b) calculate the correlation coefficient​ r, and​ (c) make a conclusion about the type of correlation. The ages​ (in years) of 6 children and the number of words in their vocabulary ​ Age, x 1 2 3 4 5 6

Vocabulary​ size, y 150 1100 1150 1800 2050 2700

A] The correlation coefficient r is

A private college report contains these​ statistics:

75​% of incoming freshmen attended public schools.

65​% of public school students who enroll as freshmen eventually graduate.

80​% of other freshmen eventually graduate.

What percent of students who graduate from the college attended a public high​ school?

___ ​% of students who graduate attended a public high school.

​(Round to two decimal places as​ needed.)

2. The Dean of Students at the University of Waterloo wanted to estimate the proportion of students who are willing to report cheating by fellow students. So, her staff surveyed the 172 students currently enrolled in the introduction to biology class. The students were asked, “Imagine that you witness two students cheating on a quiz. Would you tell the professor?” 19 of the surveyed students responded “yes.” (11 points total)
a. Using these data, calculate the 90% confidence interval for the proportion of all students at the University of Waterloo that would report cheating. (4 points)
b. Interpret the confidence interval from part “a” in a sentence. Interpret in terms of percentages, rather than proportions. (4 points)
c. Is it appropriate to use these data to estimate the proportion of all students at the university that would report cheating? Why or why not? (3 points)

You don't need to be rich to buy a few shares in a mutual fund. The question is, how reliable are mutual funds as investments? This depends on the type of fund you buy. The following data are based on information taken from a mutual fund guide available in most libraries.

A random sample of percentage annual returns for mutual funds holding stocks in aggressive-growth small companies is shown below.

Use a calculator to verify that s² ≈ 349.621 for the sample of aggressive-growth small company funds.

Another random sample of percentage annual returns for mutual funds holding value (i.e., market underpriced) stocks in large companies is shown below.

Use a calculator to verify that s² ≈ 136.311 for value stocks in large companies.

Test the claim that the population variance for mutual funds holding aggressive-growth in small companies is larger than the population variance for mutual funds holding value stocks in large companies. Use a 5% level of significance. How could your test conclusion relate to the question of reliability of returns for each type of mutual fund?

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²    H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²H_o: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

(b) Find the value of the sample F statistic. (Use 2 decimal places.)


What are the degrees of freedom?

What assumptions are you making about the original distribution?

The populations follow independent normal distributions.The populations follow dependent normal distributions. We have random samples from each population.    The populations follow independent normal distributions. We have random samples from each population.The populations follow independent chi-square distributions. We have random samples from each population.

(c) Find or estimate the P-value of the sample test statistic. (Use 4 decimal places.)

p-value > 0.1000.050 < p-value < 0.100    0.025 < p-value < 0.0500.010 < p-value < 0.0250.001 < p-value < 0.010p-value < 0.001

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

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

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

Fail to reject the null hypothesis, there is sufficient evidence that the variance in percentage annual returns for mutual funds is greater in the aggressive-growth in small companies.

Reject the null hypothesis, there is insufficient evidence that the variance in percentage annual returns for mutual funds is greater in the aggressive-growth in small companies.    

Reject the null hypothesis, there is sufficient evidence that the variance in percentage annual returns for mutual funds is greater in the aggressive-growth in small companies.

Fail to reject the null hypothesis, there is insufficient evidence that the variance in percentage annual returns for mutual funds is greater in the aggressive-growth in small companies.

The director of the Wisconsin Department of Business Licensing is looking for ways to improve employee productivity. Specifically, she would like to see an improvement in the percentage of applications that employees process correctly. The director randomly selects 50 employees and gather data on the percentage of applications each one correctly processed last month. On the recommendation of a consultant, the director has these 50 employees complete a 3-day workshop on Proactive Synergy Restructuring Techniques. At the end of the month following the training, the director collects the application processing data for the same 50 employees.

Help the director analyze these data by conducting a hypothesis test. From a statistical point of view, what can you tell the director?