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Normal Probability, Confidence Interval & Hypothesis Testing

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1- It has been reported that the mean income of parents of freshmen entering a particular university is $91,600. The president of a neighboring university feels that the mean income of the parents of his university's freshman class have a mean income greter than $91,600. The president selects 100 families randomly and finds the mean income to be $96,321 with a standard deviation of $9,555.
a) When analyzing this hypothesis, is this a two-tailed, right-tailed, or left-tailed test?
b) Using the results of this sample and a level of significance of .05, can we conclude that the he president of this neighboring university is correct in his feelings?

2- The length of human pregnancies is approximately normally distributed with a population mean of 266 days and a population standard deviation of 16 days.
a) What is the probability that a randomly selected pregnancy last less than 260 days?
b) What is the probability that a random sample of 20 pregnancies has a mean gestation period of less than 260 days?

3- The time for a Greased Lightnig oil change on a car is approximately normally distributed with an average of 17 minutes and a standard deviation or 2.5 minutes.
a) The "greased working in the pits guarantee that the oil change will take no longer than 20 minutes. If it takes longer than that, a customer will only have to pay half-fare. What is the approximate percentage of customers that can expect to pay half price?
b) If the Greased Lightning owner, Michael Blount (Chief Greaser), does not want to give this half fare price to more than 3% of its customers, how long should the company make the guarantee time limit?

4- A simple random sample of size "n" is drawn from a population that is normally distributed with a population standard deviation that is known to be 13. The sample mean, X, is found to be 108.
a) Construct a 95% CI using this data, giving the range in which the population mean is likely to be found if a sample of 25 elements is taken. (Again, show the data and the range, but you do not have to effect the full calculations.)
b) If the population standard deviation were not known, what can be used in its place (assuming we know that the population is normally distributed)?
c) If the population standard deviation is not known, what is the minimum size of the sample needed in order to use the normal distribution principles?

5 An arborist is interested in determining the mean diameter of mature white oak trees. He only has access to 7 trees. Using this sample, he determines that the sample mean is 49.09 cm and the sample standard deviation ("s") is 13.80.
Construct a 95% CI for the mean diameter of mature white oak trees using the above data.

6- For normal distributions, it is customary to contruct CI's at the 90%, 95%, and 99% levels. If we wanted to construct a CI at the 80% level, what would the z n multiplier be?

7- A two-lane highway with a posted speed limit of 45 miles per hour is located just outside a small hosing development. The residents are worried about the speed of the cards on the highway and want an estimate of the population mean speed of the cars on the highway. The transportation department of the township estimates that approximately 2500 cars traverse the highway daily during non-rush hours(when the speeds can be highter due to a lack of congestion).
This population is assumed to be normally distributed, with a population standard deviation ofo highway speed to be 8 miles per hour.
a) The township has the local high school take a sample of 12 cars during non-rush hour, selecting every 100th car until 12 cars had been measured.. The speeds measured were:
57.4 56.1 70.3 65.6
44.2 69.6 66.1 57.3
62.2 60.4 64.5 52.7
What is the point estimate of the population mean speed, using this sample data (a point estimate is the average of the numbers)?
b) Since we know that this point estimate is based on only one sample of 12 cars, we would like to find a range of speeds within which the true population mean is likely to fall. Construct a 90% confidence interval (CI) for this population mean. (do the calculation and the construction of the range of values.)
c) What is the maximum error of the mean for this CI (Calculate the number).
d) If the township were uncomfortable with this range of values, what would the multiplier be to increase the level of confidence to 99%.
e) For the 90% CI (part "b" above), what should the sample size be to limit the maximum error to within 2 miles per hour?

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Solution Summary

The solution provides step by step method for the calculation of normal probabilities, confidence intervals and testing of hypothesis. 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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Normal Probability, Confidence interval & Hypothesis Testing

I. Matching
a. Nominal _____1. Number of pages in a book
b. Ordinal _____2. Name of a book
c. Discrete _____3. Weight of a book
d. Continuous _____4. Rank on bestsellers list

II. Which of the following is not like the others in the group and say why:
mean, standard deviation, median, mode, and midrange

III for a set of ordered pairs the prediction line has b0 = 1 and b1 = 2.
We also have that one of the ordered pairs is (7, 16)
1. compute the fit
2. compute the residual

IV. Does a strong correlation prove cause and effect or not?

V. Suppose b1 is negative. What can we conclude about r
1. It is negative.
2. it is 0
3. it is positive
4. none of the above

VI. If P(A) = 0.4, P(B) = 0.3, and P(A and B) = 0.15, what is P(A|B)?

VII. You may make a Venn diagram or a table or both, but SHOW YOUR WORK
You will be hoping for a merit scholarship (M) or an athletic scholarship (A).
They say the chance of getting an athletic scholarship is 0.2
Then they say the chance of getting both an athletic scholarship and a merit scholarship is 0.05
Then they say the chance of getting one or the other is 0.5.

a. P(M) =
b. P(not A and not M) =
c. P(M|A) = (I want the ratio)
d). P(A|M) = (I want the ratio)
e. Are A and M independent? Show why you say so

VIII If A and B are mutually exclusive events with P(A) = .0.40, then P(B)
a. can be any value between 0 and 1
b. cannot be larger than 0.40
c. cannot be larger than 0.60
d. cannot be determined

IX what is the mean of the standard normal distribution (the z scores)?

X what is the approximate area under the curve between μ - 3σ and μ + 3σ?

XI what is the area under the standard normal curve between -2.41 and - 1.24?

XII Find P(z > -0.24)

XIII what z score corresponds to the 80th percentile?

XIV. If the mean of a normal distribution is 300 and the standard deviation is 50, what percentage of the scores is between 270 and 320?

XV .If it is known that college students sleep an average of 6 hours per night with a standard deviation of 1.8
1. find the probability that she sleeps between 5 and 8 hours.
2. Find the probability that she sleeps more than 7 hours

1.As the sample size increases, which happens, the sampling distribution gets more or less peaked in the middle?
2. As the sample size increases, which happens, the tails get larger or the tails get smaller?

XVII consider the confidence intervals to be alike in every way except the part I say changes and a change for part 1 does not apply for part 2
1. which is longer, a confidence interval for sample size n = 16 or n = 25?
2. which is longer, a confidence interval for 90% confidence or for 99% confidence?

XVIII. when we talk about a 95% confidence interval, what does that mean? 95% of what do what?

XIX A soft drink machine is set to dispense soft drink labeled 16 oz. If the mean is 16.1 oz with a standard deviation of 0.0.15 oz
1. what are the upper and lower limits for the 90% confidence interval for 1 bottle?
2. what are the upper and lower limits for the 90% confidence interval for a sample of size n = 20?

XX. Will a confidence interval for μ always contain the point estimate or not?

XXI If you have rejected the null hypothesis when it is false, then you have made
a. a type A correct decision
b. a type B correct decision
c. A type I error
d. A type II error

1. In a hypothesis test, if α = 0.01 and p-value = 0.019, do you have significant evidence to reject the null hypothesis or not?
2. in a hypothesis test, if α = 0.05 and the p-value = 0.042, do you have significant evidence to reject the null hypothesis?
3. Do large or small values for the p-value support the alternative hypothesis?

XXIII. Give the null and alternative hypotheses for the two tail test for the following claim "the mean of the ACT scores is 25

XXIV Which t-distribution is most like the normal curve, the one for df = 10 or n = 20?

XXV which two T VALUES bracket 2.50 on the df = 15 distribution?

XXVI pretest versus post test (before and after studies) are usually used for
a. dependent samples
b. independent samples
c. either one
d. neither applies

XXVII. consider the following set of paired data and calculate the value of A - B
A 1 3 4 4 5
B 3 5 4 1 2
A - B

XXVIII If a confidence interval for the difference of 2 proportions contains both positive and negative numbers, what does that tell you?

XXIX Only 48 out of 200 people interviewed were able to name the Secretary of State of the United States. Find the values of
x =
n =

XXX Here are expected values for a chisquared χ2 test
41 44
72 71

1. Compute the expected value for the cell in the first column, first row
2. show the formula and plug in the numbers for computing the addend for chisquared for the first column and the first row.
3.how many degrees of freedom does this test have?

4. If we had the chisquared sum = 24 with df = 15, WHICH TWO VALUES FROM THE TABLE WOULD BRACKET THE P VALUE?

XXXI Can we use an ANOVA test to see whether our linear model is good in regression? There is no work to show, but tell me why you think you are right.

**Real ANOVA question

I give you the df and SS numbers. I want the MS numbers and the F number. You don't have enough information to find the p-value
Source df SS MS F
Model 2 29.20
Error 12 136.13

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