The scientific productivity of major world cities was the subject of a recent study. The study determined the number of scientific papers published between 1994 and 1997 by researchers from each of the 20 world cities, and is shown below.

Construct a 99 % confidence interval for the average number of papers published in major world cities.

<μ<

7.46 A poll had a sample of 66 people choose their favorite Skittles flavor by color (green, orange, purple, red or yellow). A separate poll sampled 91 people, again asking them their favorite Skittles flavor, but rather than by color they asked by the actual flavor (lime, orange, grape, strawberry, and lemon, respectively). The table below shows the results from both polls. Does the way people choose their favorite Skittles type, by color or flavor, appear to be related to which type is chosen?

1. State the null and alternative hypotheses.
2. Give a table with the expected counts for each of the 10 cells.
3. Are the expected counts large enough for a chi-square test?
4. How many degrees of freedom do we have for this test?
5. Calculate the chi-square test statistic.
6. At the α=0.05 level, do we find evidence that method of choice affects which is chosen?

Table: Skittles popularity

Please answer using your own words (i.e., don't quote directly from the text).: Describe and contrast the three ways of describing results: comparing group percentages, correlating scores, and comparing group means. For each, give two examples (other than those given in the text) of their appropriate use.

What is a z-test, when is it used, and what does it tell us?

A criminologist conducted a survey to determine whether the incidence of certain types of crime varied from one district of a large city to another. The particular crimes of interest were assault, burglary, larceny, and homicide. The following table shows the number of crimes committed in four districts of the city during the past year.

Type of Crime

Can we conclude from the data at the 0.01 significance level that the occurrence of these types of crime is dependent on the city district?

Can we conclude from the data at the 0.01 significance level that the occurrence of these types of crime is dependent on the city district?

       Note:   The confidence level is equivalent to 1 – α. So, if your significance

                    level is 0.05, the corresponding confidence level is 95%.

Note: You can use the functions qchisq() in R to help you in solving the following.

Why we are using qchisq() function in R?

The qchisq() function in R allows us to specify a desired area in a tail and the number of degrees of freedom. From that information, qchisq() computes the required x-value to get the specified area in the specified tail with the specified number of degrees of freedom.

1. (1 point) State the two hypothesis of interest.

2. (2 points) Calculate an appropriate test statistic for (a) by hand. Use the table below to do the calculations of the expected frequencies.

Note: Round the numbers to two decimals.

C. (2 points) Write your conclusion using the rejection region method “critical value method” include both statistical and related to the topic of the question (practical) interpretation use the function qchisq() in R

A researcher studied the relationship between the salary of a working woman with school-aged children and the number of children she had. The results are shown in the following frequency table:

                                                Number of Children

          If a working woman has more than 2 children, what is the probability she has a low or medium salary?

          A.       0.79            B.      0.45            C.      0.35            D.      0.95

14.     The expected number of heads in 410 tosses of a fair coin is:

1.    205               B.       185             C.      195             D.      175

Mercury pollution is a serious ecological problem. It typically becomes dangerous once it falls into large bodies of water. At this point microorganisms change it into methylmercury (CH₃²⁰³). The fish consume these microorganisms which makes them contaminated and hence anyone eating those fish are at risk.

Because of this, investigators are interested in research around mercury poisoning. In particular they want to investigate the methlymercury metabolism and whether it proceeds at a different rate for women than for men. The table below captures the half-life (in days) of an oral administration of protein-bound methlymercury among six females and nine males. Round all the numbers to 2 decimal places.

1. (2 points) Create and interpret a 95% confidence interval for the average methylmercury half-life in males fish. Note: Do the calculations one time using R and the other time by hand.
2. (2 points) Create and interpret a 95% confidence interval for the average methylmercury half-life in females. Note: Do the calculations one time using R and the other time by hand.
3. (2 points) In comparing the confidence intervals created in (a) and (b) does there appear to be a difference in the average methylmercury half-life between males and females? Explain.

MUST SHOW HOW TO DO IT IN R AND BY HAND

A population has a mean of 200 and a standard deviation of 90. Suppose a sample of size 125 is selected x_bar and is used to estimate μ Use z-table.

a. What is the probability that the sample mean will be within +/- 7 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.)

b. What is the probability that the sample mean will be within +/- 15 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.)

“Suppose you are an educational researcher who wants to increase the science test scores of high school students. Based on tremendous amounts of previous research, you know that the national average test score for all senior high school students in the United States is 50 with a standard deviation of 20.

“Write H0 next to the verbal description of the null hypothesis and H1 next to the research hypothesis.
_____The population of students who receive tutoring will have a mean science test score that is equal to 50.
_____The population of students who receive tutoring will have a mean science test score that is greater than 50.
_____The population of students who receive tutoring will not have a mean science test score that is greater than 50.
_____The population of students who receive tutoring will have a mean science test score that is less than 50.”

A factor in determining the usefulness of an examination as a measure of demonstrated ability is the amount of spread that occurs in the grades. If the spread or variation of examination scores is very small, it usually means that the examination was either too hard or too easy. However, if the variance of scores is moderately large, then there is a definite difference in scores between "better," "average," and "poorer" students. A group of attorneys in a Midwest state has been given the task of making up this year's bar examination for the state. The examination has 500 total possible points, and from the history of past examinations, it is known that a standard deviation of around 60 points is desirable. Of course, too large or too small a standard deviation is not good. The attorneys want to test their examination to see how good it is. A preliminary version of the examination (with slight modifications to protect the integrity of the real examination) is given to a random sample of 20 newly graduated law students. Their scores give a sample standard deviation of 64 points. Using a 0.01 level of significance, test the claim that the population standard deviation for the new examination is 60 against the claim that the population standard deviation is different from 60.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: σ = 60; H₁: σ > 60H_o: σ = 60; H₁: σ < 60    H_o: σ > 60; H₁: σ = 60H_o: σ = 60; H₁: σ ≠ 60

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume a exponential population distribution.We assume a binomial population distribution.    We assume a normal population distribution.We assume a uniform population distribution.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-value < 0.005

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

Since the P-value > α, we fail to reject the null hypothesis.Since the P-value > α, we reject the null hypothesis.    Since the P-value ≤ α, we reject the null hypothesis.Since the P-value ≤ α, we fail to reject the null hypothesis.

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

At the 1% level of significance, there is insufficient evidence to conclude that the standard deviation of test scores on the preliminary exam is different from 60.At the 1% level of significance, there is sufficient evidence to conclude that the standard deviation of test scores on the preliminary exam is different from 60.    

(f) Find a 99% confidence interval for the population variance. (Round your answers to two decimal places.)

(g) Find a 99% confidence interval for the population standard deviation. (Round your answers to two decimal places.)

An article describes an experiment to determine the effectiveness of mushroom compost in removing petroleum contaminants from soil. Out of 155 seeds planted in soil containing 3% mushroom compost by weight, 74 germinated. Out of 155 seeds planted in soil containing 5% mushroom compost by weight, 86 germinated. Can you conclude that the proportion of seeds that germinate differs with the percent of mushroom compost in the soil? Find the P-value and state a conclusion.

The National Institute on Alcohol Abuse and Alcoholism defines binge drinking as a pattern of drinking that brings blood alcohol concentration (BAC) levels to 0.08g/dL. It is cited as the most common and deadly pattern of alcohol abuse in the country, which can cause many health problems such as alcohol poisoning, sudden infant death syndrome, and chronic diseases, to name a few. In the binge drinking fact sheet published by the Center for Disease Control and Prevention, the amount of binge drinks consumed per year by binge drinkers are greater among those with lower incomes (below $75000) and educational level. In order to verify if this claim is true, a random sample of binge drinkers from the two income groups were obtained, and the data are summarized in the table below:

Conduct a test of hypothesis at 5% level of significance to verify the claim.

What is your conclusion in the context of the problem?

What is the income distribution of super shoppers? A supermarket super shopper is defined as a shopper for whom at least 70% of the items purchased were on sale or purchased with a coupon. In the following table, income units are in thousands of dollars, and each interval goes up to but does not include the given high value. The midpoints are given to the nearest thousand dollars. Income range 5-15 15-25 25-35 35-45 45-55 55 or more Midpoint x 10 20 30 40 50 60 Percent of super shoppers 20% 14% 22% 17% 20% 7%

A red and a green die are rolled. Chart or graph the sample space, and find the odds that the numbers on the dice differ by 1 or more

In a random sample of 85 automobile engine crankshaft bearings, 10 have a surface finish that is rougher than the specifications allow.

1. Construct a 95% two-sided confidence interval for the true proportion of bearings in the population that exceeds the roughness specification.

2. How large a sample is required if we want to be 95% confident that the error in using the sample proportion to estimating the ture value p is less than 5%?

3. How large must the sample be if we wish to be at least 95% confident that the error in estimating the true proportion is less than 5% regardless of the true value of P ?