Question 2
Raw data: 19 14 25 17 29 24 36 23 9 26 22 31 19 28 8
2.1 Group the data into a frequency distribution with a lowest class lower limit of 8 and class width of 7, then draw an ogive curve and use it to estimate the mean
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General guidelines:
Use EXCEL or PHStat to do the necessary computer work.
Do all the necessary analysis and hypothesis test constructions, and explain completely.
Read the textbook Chapter 13. Imagine that you are managing a mobile phone company. You want to construct a simple linear regression model to capture and represent the relationship between the number of customers and the annual sales level for a year with 95% confidence. You had conducted a pilot study for the past fifteen years and collected yearly observations as given in the following data.Where the number of customers in a year is represented by the Profiled Customers variable, measured by million customers unit, and the sales level is represented by the Annual Sales variable, measured by million US-dollars unit.
1) Investigate the agreement between the model and the data set for:
A) LINEARITY.
A1) Construct the "Dot Plot", a.k.a. "Scatter Plot," for this data. Visually inspect for the linear relationship between the number of customers and the sales level. Make comments based on your observations.
A2) Conduct the F-Test for linearity.
A3) If you have seen evidence of linearity in the F-Test, then:
Conduct the t-Test for the partial slope.
Construct the 95% Confidence Interval Estimator for the partial slope.
Thus, make comments about the linear relationship between the Profiled Customers and the Annual Sales, based on the partial slope information.
B) NORMALITY.
Construct the "Normal Probability Plot" for the Annual Sales variable, and make comments about the normality of annual sales level, based on your observations.
C) HOMOSCEDASTICITY.
Construct the "Residual Plot" and make comments about the variance of annual sales level, based on your observations.
D) INDEPENDENCE.
This data set is a Time-Series. Hence, investigate for the independence of observations in this time-series, based on the Durbin-Watson test.
2) If there is evidence of agreement between the model and data, and independence of observations, then construct the simple linear regression equation for this data set, based on the least square error method.
2A) Construct the 95% confidence interval for the actual average annual sales level for all the years that you have 5 million customer in a year,
2B) Construct the 95% prediction interval for the actual annual sales level for one year that you have 5 million customers in that year.
| Years | Profiled Customers | Annual Sales | |
| 1 | 3.7 | 5.7 | |
| 2 | 3.6 | 5.9 | |
| 3 | 2.8 | 6.7 | |
| 4 | 5.6 | 9.5 | |
| 5 | 3.3 | 5.4 | |
| 6 | 2.2 | 3.5 | |
| 7 | 3.3 | 6.2 | |
| 8 | 3.1 | 4.7 | |
| 9 | 3.2 | 6.1 | |
| 10 | 3.5 | 4.9 | |
| 11 | 5.2 | 10.7 | |
| 12 | 4.6 | 7.6 | |
| 13 | 5.8 | 11.8 | |
| 14 | 2.9 | 4.1 | |
| 15 | 3 | 4.1 |
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An SAT prep course claims to improve the test score of students. The table below shows the scores for seven students the first two times they took the verbal SAT. Before taking the SAT for the second time, each student took a course to try to improve his or her verbal SAT scores. Do these results support the claim that the SAT prep course improves the students' verbal SAT scores?
Let d=(verbal SAT scores prior to taking the prep course)−(verbal SAT scores after taking the prep course)d=(verbal SAT scores prior to taking the prep course)−(verbal SAT scores after taking the prep course). Use a significance level of α=0.1 for the test. Assume that the verbal SAT scores are normally distributed for the population of students both before and after taking the SAT prep course.
| Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Score on first SAT | 400 | 420 | 510 | 530 | 380 | 440 | 460 |
| Score on second SAT | 440 | 490 | 560 | 560 | 410 | 510 | 500 |
Step 1 of 5: State the null and alternative hypotheses for the test.
Step 2 of 5: Find the value of the standard deviation of the paired differences. Round your answer to one decimal place.
Step 3 of 5: Compute the value of the test statistic. Round your answer to three decimal places.
Step 4 of 5: Determine the decision rule for rejecting the null hypothesis H 0 . Round the numerical portion of your answer to three decimal places.
Step 5 of 5: Make the decision for the hypothesis test.
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The American Association of Individual Investors (AAII) On-Line Discount Broker Survey polls members on their experiences with electronic trades handled by discount brokers. As part of the survey, members were asked to rate their satisfaction with the trade price and the speed of execution, as well as provide an overall satisfaction rating. Possible responses (scores) were no opinion (0), unsatisfied (1), somewhat satisfied (2), satisfied (3), and very satisfied (4). For each broker, summary scores were computed by computing a weighted average of the scores provided by each respondent. A portion the survey results follow (AAII website, February 7, 2012).
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The population average cholesterol content of a certain brand of egg is 215 milligrams, and the standard deviation is 15 milligrams. Assume the variable is normally distributed.
i) Find the third quartile for the average cholesterol content for 25 eggs.
ii)If we are told the average for 25 eggs is less than 220 mg, what is the probability that the average is less than 210.
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American football is the highest paying sport on a per-game basis. The quarterback, considered the most important player on the team, is appropriately compensated. A sports statistician wants to use 2009 data to estimate a multiple linear regression model that links the quarterback’s salary (in $ millions) with his pass completion percentage (PCT), total touchdowns scored (TD), and his age. A portion of the data is shown in the accompanying table. Name Salary PCT TD Age Philip Rivers 25.5566 65.2 28 27 Jay Cutler 22.0441 60.5 27 26 ⋮ ⋮ ⋮ ⋮ ⋮ Tony Romo 0.6260 63.1 26 29
c. Drew Brees earned 12.9895 million dollars in 2009. What is his predicted salary if PCT = 70.6, TD = 34, and Age = 30? (Round coefficient estimates to at least 4 decimal places and final answer to 2 decimal places.) Salaryˆ
d. Tom Brady earned 8.0073 million dollars in 2009. According to the model, what is his predicted salary if PCT = 65.7, TD = 28, and Age = 32? (Round coefficient estimates to at least 4 decimal places and final answer to 2 decimal places.) Salaryˆ e-1. Compute the residual salary for Drew Brees and Tom Brady. (Negative values should be indicated by a minus sign. Round coefficient estimates to at least 4 decimal places and final answer to 2 decimal places.)
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In problems 1 – 5, a binomial experiment is conducted with the given parameters. Compute the probability of X successes in the n independent trials of the experiment.
1. n = 10, p = 0.4, X = 3
2. n = 40, p = 0.9, X = 38
3. n = 8, p = 0.8, X = 3
4. n = 9, p = 0.2, X < 3
5. n = 7, p = 0.5, X = > 3
According to American Airlines, its flight 1669 from Newark to Charlotte is on time 90% of the time. Suppose 15 flight are randomly selected and the number of on – time flights is recorded.
a. Find the probability that exactly 14 flights are on time.
b. Find the probability that at least 14 flights are on time.
c. Find the probability that fewer than 14 flights are on time.
d. Find the probability that between 12 and 14 flights are on time.
e. Find the probability that every flight is on time.
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The quarterly returns for a group of 53 mutual funds with a mean of 2.1% and a standard deviation of 5.1% can be modeled by a Normal model. Based on the model N(0.021,0.051), what are the cutoff values for the
a) highest 10% of these funds?
b) lowest 20%?
c) middle 40%?
d) highest 80%?
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Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean u=266 and standard deviation 26 days. (a) what is the probability that a randomly selected pregnancy lasts less than 256 days? (b) what is the probability that a random sample of 16 preganacies has a mean geatation period of 256 days or less? (c) what is the probability that a random sample of 37 preganacies has a mean geatation period of 256 days or less? MUST SHOW WORK
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Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 130 millimeters, and a standard deviation of 6 millimeters. If a random sample of 33 steel bolts is selected, what is the probability that the sample mean would be greater than 130.6 millimeters? Round your answer to four decimal places.
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According to Nielsen Media Research, the average number of hours of TV viewing by adults (18 and over) per week in the United States is 36.07 hours. Suppose the standard deviation is 9.7 hours and a random sample of 52 adults is taken.
a. What is the probability that the sample average is more than 35 hours?
b. What is the probability that the sample average is less than 38.8 hours?
c. What is the probability that the sample average is less than 29 hours? If the sample average actually is less than 40 hours, what would it mean in terms of the Nielsen Media Research figures?
d. Suppose the population standard deviation is unknown. If 75% of all sample means are greater than 48 hours and the population mean is still 36.07 hours, what is the value of the population standard deviation?
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The contents of bottles of beer are Normally distributed with a mean of 300 ml and a standard deviation of 5 ml.
What is the probability that the average contents of a six-pack will be between 293 ml and 307 ml?
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1. Murphy’s Law, a pub in downtown Rochester, claims its patrons average 25 years of age. A random sample of 40 bar patrons is taken and their mean age is found to be 26.6 years with a standard deviation of 4.5 years. Do we have enough evidence to conclude at the level of significance α = 0.025 that, on average, patrons at Murphy’s Law are older than 25?
Determine the test statistic.
Determine the range of P-values.
Write your decision and explain how you reached it.
Write the conclusion that addresses the original claim.
Determine the type of error you could have made and explain why. (Type I or Type II
error)
2. The coffee machine at your company has been acting weirdly these past few days. You were assured that the machine would pour 7 ounces of coffee every time you press the button. You believe that this is not true anymore and want to call the technician. Before you do so, you gather a sample of 15 coffees and find out that the mean amount of coffee in each cup is 6.8 ounces with a standard deviation of 1 ounce. Is there sufficient evidence to show that the population mean is different than 7 ounces? Perform a test at the 0.05 level of significance. Assume normality.
Determine the test statistic.
Determine the range of P-values.
Write your decision and explain how you reached it.
Write the conclusion that addresses the original claim.
Determine the type of error you could have made and explain why. (Type I or Type II
error)
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A well-designed questionnaire should meet the research objectives. Give examples of preparatory work that one should conduct to ensure that these objectives are met.
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The age distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of a village are shown below.
| Age (years) | Percent of Canadian Population | Observed Number in the Village |
| Under 5 | 7.2% | 47 |
| 5 to 14 | 13.6% | 72 |
| 15 to 64 | 67.1% | 295 |
| 65 and older | 12.1% | 41 |
Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.
(a) What is the level of significance?
State the null and alternate hypotheses.
H0: The distributions are different.
H1: The distributions are
different.H0: The distributions are the
same.
H1: The distributions are the
same. H0: The
distributions are different.
H1: The distributions are the
same.H0: The distributions are the same.
H1: The distributions are different.
(b) Find the value of the chi-square statistic for the sample.
(Round your answer to three decimal places.)
Are all the expected frequencies greater than 5?
YesNo
What sampling distribution will you use?
uniformStudent's t binomialnormalchi-square
What are the degrees of freedom?
(c) 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 that the population fits the
specified distribution of categories?
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 5% level of significance, the evidence is insufficient to conclude that the village population does not fit the general Canadian population.At the 5% level of significance, the evidence is sufficient to conclude that the village population does not fit the general Canadian population.
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