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Hypothesis Testing & Probability

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1- 20% of all Toyota's are painted red. Using a car lot of Toyota's reflecting this relationship,
a) If I select 3 toyota's in succession, what is the probability that all will be red?
b) If I select 3 toyota's in succession, what is the probability that none is red?
c) If I select 3 toyota's in succession, what is the probability that at least one is red?
d) Check your answers using the Binomial Distribution table and the above questions. (Hint: "n" =3, p=.2, q=.8 and you set "x" to the required number).
2- Given the following set of numeric variables. Construct a discrete probability distribution reflecting the pobability of selecting any of the variables. You can us fractions to represent the probabilities. [1,1,3,5,5,7,7,9,11]
x -----------------------------------------------------------------
P(X)
a) What is the probability of selecting a single variable and getting a "1?"
b) What is the probability of selecting a single variable and getting a"3" or a "5?"
c) What is the mean of this probability distribution?
d) What is the expected alue of this distribution?
e) Calculate the mean of the set of variables from the formula "Ex / n." Does it agree with your answer to "c" above?
f) Calculate the variance and standard deviation for the above distribution.
g) Cna you use Chebyshev's Theorem to calculate the percentage of the data falling within 1.5 standard deviations of this mean? State the formula to be used. What is the restriction associated with the value of the standard deviation?
3- A thermometer company manufactures thermometers that are supposed to give readings of 0(degree) C at the freezing point of water. A series of tests of these readings yielded a mean of 0(degree) C with a standard deviation of 1.0(degree) C. The readings were normally distributed. If one thermometer is selected randomly from the manufacturing output, what is the probality that:
a) At the freezing pint of water the reading is less han 1.58(degree) C? (Draw your picture)
b) A randomly selected thermometer will read above 1.23(degree) C at the freezing point?
c) A randomly selected thermometer reads between -2.00(degree) C and +1.5(degree) C at the freezing point?
4- State the standard formula for a Confidence Interval.
a) What is "X?"
b) What does the symbol "a" represent?
c) What are the multipiers for the 90%m 95%, and 99% CI's?
d) Quincy College wants to estimate the estimate the average age ("u") of its students from a sample of 50 students. It is known that the population standard deviation is 2 years. The mean of the sample selected is 23.2 years.
Develop the 95% CI within which the population mean will fall.
e) For this same Quincy College study, I want to estmate te averae age of the student population with a 99% CI level accurate within 1 year (i.e., the maximum error of the range is to be +/- year. From the information above, the standard deviation was stated to be 2 years.
What should the size of the sample be in order to limit the maximum error to 1 year?
5- Birth weights in the U.S. are normally distributed with a mean of 3,420g and a standard deviation of 495g "g" means "gram").What weight is the cut-off weight separating the highest 2% of babies (in terms of weights) from the res of the population of babies?
6- Hypthesis Testing Basics. I have built a new Roman Candle that goes higher than the existing Roman Candles that are sold. The mean height of today's existing Roman Candles that are sold. The mean height of today's existing Roman Candles is 75 feet.
a) State the null and alternative hypoteses. Ho: H1:
b) Is this a 2-tailed,left-tailed, or a right-tailed test/
c) If I want to test my new process with a level of ignificance of .05, draw a picture of the region where the mean of the sample used for testing must fall for me to be able to reject the null hypothesis.
7- Sample size problem. I want to find the true population mean cost of a large pizza with a 95% level o confidence. (The (sigma) of the population is \$0.26.) How large should the sample be if I want the cost to be accurate within \$0.15?
8- I have a four-sidd figure like a die (but only 4 sides) with the numbers {1,1,3,7}. If I roll the figure twice:
a) Draw a tree diagram showing the possible outcomes.
b) What is the probability that I roll two 3's on the two rolls?
9- 20% of the population has a recessive gene that allows them to cross their eyes and wiggle their ears at the same time. If we choose a group of 10 individuals, what is the probability that: a) Exactly 2 have this capability?
b) No more than 3 individuals exhibit this trait?
c) At least 4 exhibit this trait/
d) Calculate the mean, variance, and standard deviation of this distribution.
10- A survey of the wights of men yield a normal distribution with a mean of 172 lbs and a standard deviation of 29 lbs.
a) Find the probability that an individual man randomly selected will have a weight greater than 175 lbs? (Ddraw your picture).
b) If I select a group of 20 individuals, what is the probability that the mean of the sample will be greater than 175 lbs?
c) When working with samples: 1) What is he adjustment that must be made to the population standard deviation in order to calculate a normally distributed probability of the mean of the sample?
2) In order to use the normal distribution tale for probabilities, what are the two criteria that must be met (only one or the othr is required?
11- An actual hypothesis test: The average salary of Quincy College professors is \$42,000 with a (sigma) of \$5,230. I selected a sample of 50 professors and obtain mean salary of \$43,260 for this sample. With a level of significance of .05, test the hypothesis that Quincy College professors earn more tha \$42,000,
a) State the null and alternative hypotheses:
Ho: H1:
b) Is this a two-tailed test, a right-tailed test, or a left-tailed test?
c) Draw a picture of the distibution, shading in the critical region.
d) Using the graph (picture) above, plot the test value on the graph.
e) Can we reject Ho?

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Solution Summary

The solution provides step by step method for the calculation of testing of hypothesis and probability. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

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Franklin Park Mall Hypothesis Testing

The owners of the Franklin Park Mall wished to study customer shopping habits. From earlier
studies the owners are under the impression that a typical shopper spends 0.75 hours at the mall,
with a standard deviation of 0.10 hours. Recently the mall owners added some specialty restaurants designed to keep shoppers in the mall longer. The consulting firm, Brunner and Swanson Marketing Enterprises, has been hired to evaluate the effects of the restaurants. A sample of 45 shoppers by Brunner and Swanson revealed that the mean time spent in the mall had increased to 0.80 hours.
a. Develop a test of hypothesis to determine if the mean time spent in the mall is more than
0.75 hours. Use the .05 significance level.
b. Suppose the mean shopping time actually increased from 0.75 hours to 0.77 hours. What is
the probability this increase would not be detected?
c. When Brunner and Swanson reported the information in part (b) to the mall owners, the owners
were upset with the statement that a survey could not detect a change from 0.75 to 0.77
hours of shopping time. How could this probability be reduced?

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