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Questions and Answers of Statistics

List all 8 sub sets of the sample space S = {a, b, c}.
Using Venn diagrams graph and check the rules
Using a Venn diagram, show that A C B if and only if A U B = B.
In Prob. 1 find the probability of E: At least 1 defective (i) directly, (ii) by using complements; in both cases (a) and (b).
Under what conditions will it make practically no difference whether we sample with or without replacement? Give numeric examples.
If in Prob. 5 you draw without replacement, does the probability decrease or increase? First think then calculate.
Suppose that we draw cards repeatedly and with replacement from a file of 100 cards, 50 of which refer to male and 50 to female persons. What is the probability of obtaining the second "female" card
In rolling two fair dice, what is the probability of obtaining a sum greater than 4 but not exceeding 7?
Solve Prob. 11 by considering complements
If a circuit contains 3 automatic switches and we want that, with a probability of 95%, during a given time interval they are all working, what probability of failure per time interval can we admint
In Prob. 15, what is the probability that at least one of the tires will not last for 25 000 miles?
Show that if B is a subset of A, then O(B) < P(A).
You may wonder whether in (16) the last relation follows from the others, but the answer is no. To see this, imagine that a chip is drawn from a box containing 4 chips numbered 000, 011, 101, 110,
List(a) All permutations,(b) All combinations without repetitions,(c) All combinations with repetitions, of 5 letters a, e, i, o, u taken 2 at a time.
If a cage contains 100 mice, two of which are male, what is the probability that the two male mice will be included if 12 mice are randomly selected?
By what factor is the probability in Prob. 9 decreases if the number of balls is doubled (4 blue, etc)?
Birthday problem what is the probability that in a group of 20 people (that includes no twins) at least two have the same birthday, if we assume that the probability of having birthday on a given day
(a) Prove Theorem 2.(b) Prove the last statement of Theorem 3.(c) Derive (11) from (8)(d) By the binomial theorem,So that akbn-k has the coefficient (nk). Can you conclude this from Theorem 3 or is
Graph the density function f(x) = kx2 (0 < x < 5; k suitable) and the distribution function.
In Prob. 3 find P(0 < x < 4) and c such that P (– c < X < c) = 95%.
Graph the distribution function F(x) = 1 – e–3x if x > 0, F(x) = 0 if x < 0, and the density f(x) find x such that F(x) = 0.9
If X has the probability function f(x) = k/2∞ (x = 0, 1, 2, ∙∙∙), what are k and P(X > 4)?
Suppose that certain bolts have length L = 200 + X mm, where X is a random variable with density f(x) = ¾(1 – x2) if – 1 < x < 1 and 0 otherwise. Determine c so that with a probability of 95% a
Suppose that in an automatic process of filling oil into cans, the content of can (in gallons) is Y = 50 + X, where X is a random variable with density f(x) = 1 – |x| when |x| ≤ 1 and 0 when |x|
If the diameter X of axles has the density f(x) = k if 119.9 < x < 120.1 and 0 otherwise, how many defectives will a lot of 500 axles approximately contain if defectives are axles slimmer than 119.92
Suppose that in Prob. 13, a bolt is regarded as being defective if its diameter deviates from 1.00 cm by more than 0.09cm. What percentage of defective bolts should we then expect?
A box contains 4 right-handed and 6 left-handed screws, two screws are drawn at random without replacement. Let X be the number of left-handed screws drawn. Find the probabilities P(X = 0), P(X = 1),
What is the mean life of a light bulb whose life X (hours) has the density f(x) = 0.001e–0.001x (x > 0)?
Suppose that 3% of bolts made by a machine are defective occurring at random during production. If the bolts are packaged 50 per box, what is the Poisson approximation of the probability that a given
Suppose that a telephone switchboard of some company on the average handles 300 calls pre hours, and that the board can make at most 10 connections per minute. Using the Poisson distribution,
A process of manufacturing screws is checked every hour by inspecting n screws selected at random from that hour’s production. If on or more screws are defective, the process is halted and
A carton contains 20 fuses, 5 of which are defective. Find the probability that, if a sample of 3 fuses is chosen from the carton by random drawing without replacement, x fuses in the sample will be
Let X is normal with mean 120 and variance 16. Find P(X < 126), P(X > 116,) P (125 < X < 130).
Let X be normal with mean 4.2 and variance 0.04, find c such that P(X < c) = 50%, P(X > c) = 10%, P (1 – c < X – 4.2 < c) = 99%.
The breaking strength X [kg] of a certain type of plastic block is normally distributed with a mean of 1250kg and a standard deviation of 5.5kg, what is the maximum load such that we can expect no
If the monthly machine repair and maintaneance cost X in a certain factory is known to be normal with mean $12000 and standard deviation $2000, what is the probability that the repair cost for the
Find P(X > 2, Y > 2) and P(X < 1, Y < 1) if (X, Y) has the density f(x, y) = 1/8 if x > 0, y > 0, x + y < 4.
Find the density of the marginal distribution of X in Prob.2.
If certain sheets of wrapping paper have mean weight of 10g each, with a standard deviation of 0.05g, what are the mean weight and standard deviation of a pack or 10 000 sheets?
Let X [cm] and Y [cm] be the diameter of a pin and hole, respectively, suppose that (X, Y) has the density f(x, y) = 2500 if 0.99 < x < 1.01, 1.00 < y < 1.02 and 0 otherwise.(a) Find the marginal
Let (X, Y) have the density f(x, y) = k if x2 + y2 < 1 and 0 otherwise. Determine k find the densities of the marginal distribution find the probability P(X2 + Y2 < ¼).
Let X be the ratio of sales to profits of some firm, assume that X has the distribution function F(x) = 0 if x < 2, F(x) = (x2 – 4)/5 if 2 < x < 3, F(x) = 1 if x > 3. Find and graph the density,
What capacity must the tank in Prob. 11 have in order that the probability that the tank will be emptied in a given week be 5%?
Suppose that mi allergist wishes to test the hypothesis that at least 30% of the public is allergic to some cheese products. Explain how the allergist could commit(a) A type I error;(b) A type
A sociologist is concerned about the effectiveness of a training course designed to get more drivers to use seat, belts in automobiles.(a) What hypothesis is she testing if she commits a type I error
A large manufacturing firm is being charged with discrimination in its hiring practices.(a) What hypothesis is being tested if a jury commits a type I error by dueling the firm guilty?(b) What
The proportion of adults living in a small town who are college graduates is estimated to be p = 0.6.To test this hypothesis, a random sample of 15 adults is selected. If the number of college
Repeat Exercise 10.4 when 200 adults are selected and the fail to reject region is defined to be 110 < x < 130 where x is the number of college graduates in our sample. Use the normal
A fabric manufacturer believes that the proportion of orders for raw material arriving late is p = 0.6. If a random sample of 10 orders shows that 3 or fewer arrived late, the hypothesis that p = 0.6
Repeat Exercise 10.6 when 50 orders are selected and the critical region is defined to be x < 24, where x is the number of orders in our sample that arrived late. Use the normal approximation.
A dry cleaning establishment claims that a new spot remover will remove more than 70% of the spots to which it is applied. To check this claim, the spot remover will be used on 12 spots chosen at
Repeat Exercise 10.8 when 100 spots are treated and the critical region is defined to be x > 82, where x is the number of spots removed.
Repeat Exercise 10.8 when 100 spots are treated and the critical region is defined to be x > 82, where x is the number of spots removed. Discuss.
Repeat Exercise 10.10 when 70 patients are given the mussel extract and the critical region is defined to be x < 24, where x is the number of osteoarthritic patients who receive relief.
A random sample of 400 voters in a certain city are asked if they favor an additional 4% gasoline sales tax to provide badly needed revenues for street repairs. If more than 220 but fewer than 260
Suppose, in Exercise 10.12, we conclude that 60% of the voters favor the gasoline sales tax if more than 214 but fewer than 266 voters in our sample favor it. Show that this new critical region
A manufacturer has developed a new fishing line, which he claims has a mean breaking strength of 15 kilograms with a standard deviation of 0.5 kilogram. To test the hypothesis that, p = 15 kilograms
A soft-drink machine at a steak house is regulated so that the amount of drink dispensed is approximately normally distributed with a mean of 200 milliliters and a standard deviation of 15
Repeat Exercise 10.15 for samples of size n – 25. Use the same critical region.
A new curve has been developed for a certain type of cement that results in a compressive strength of 5000 kilograms per square centimeter and a standard deviation of 120. To test the hypothesis that
If we plot the probabilities of failing to reject Ho corresponding to various alternatives for μ (including the value specified by Ho) and connect all the points by a smooth curve, we obtain the
An electrical firm manufactures light bulbs that have a lifetime that is approximately normally distributed with a mean of 800 hours and a standard deviation of 40 hours. Test the hypothesis that p =
A random sample of 64 bags of white Cheddar popcorn weighed, on average, 5.23 ounces with a standard deviation of 0.24 ounces. Test the hypothesis that p = 5.5 ounces against the alternative
In a research report by Richard H. Weindruch of the UCLA Medical School, it is claimed that mice with an average life span of 32 months will live to be about 40 months old when 40% of the calories in
The average height of females in the freshman class of a certain college has been 162.5 centimeters with a standard deviation of 6.9 centimeters. Is there reason to believe that there has been a
It is claimed that an automobile is driven on the average more than 20,000 kilometers per year. To test this claim a random sample of 100 automobile owners are asked to keep a record of the
In the American Heart Association journal Hypertension, researchers report that individuals who practice Transcendental Meditation (TM) lower their blood pressure significantly. If a random sample of
Test the hypothesis that the average content of containers of a particular lubricant is 10 liters if the contents of a random sample of 10 containers are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4,
According to a dietary study, a high sodium intake may be related to ulcers, stomach cancer, and migraine headaches. The human requirement for salt is only 220 milligrams per day, which is surpassed
A study at the University of Colorado at Boulder shows that running increases the percent resting metabolic rate (RMR) in older women. The average RMR of 30 elderly women runners was 34.0% higher
According to Chemical Engineering an important property of fiber is its water absorbency. The average percent absorbency of 25 randomly selected pieces of cotton fiber was found to be 20 with a
Past experience indicates that the time for high school seniors to complete a standardized test is a normal random variable with a mean of 35 minutes. If a random sample of 20 high school seniors
A random sample of size n1 = 25, taken from a normal population with a standard deviation σ1 = 5.2, has a mean x1 = 81. A second random sample of size n2 = 36, taken from a different normal
A manufacturer claims that the average tensile strength of thread A exceeds the average tensile strength of thread B by at least 12 kilograms. To test his claim, 50 pieces of each type of thread are
Am stat News (December 2004) lists median salaries for associate professors of statistics at research institutions and at liberal arts and other institutions in the United States. Assume a sample of
A study is made to see if increasing the substrate concentration has an appreciable effect on the velocity of a chemical reaction. With a substrate concentration of 1.5 moles per liter, the reaction
A study was made to determine if the subject matter in a physics course is better understood when a lab constitutes part of the course. Students were randomly selected to participate in either a
To find out whether a new serum will arrest leukemia, 9 mice, all with an advanced stage of the disease, are selected. Five mice receive the treatment and 4 do not. Survival times, in years, from the
A large automobile manufacturing company is trying to decide whether to purchase brand A or brand B tires for its new models. To help arrive at a decision, an experiment is conducted using 12 of each
In Exercise 9.42 on page 298, test the hypothesis that Volkswagen mini-trucks, on the average, exceed similarly equipped Toyota mini-trucks by 4 kilometers per liter. Use a 0.10 level of significance.
A UCLA researcher claims that the average life span of mice can be extended by as much as 8 months when the calories in their food are reduced by approximately 40% from the time they are weaned. The
The following data represent the running times of films produced by 2 motion-picture companies. Test the hypothesis that the average running time of films produced by company 2 exceeds the average
In a study conducted at the Virginia Polytechnic Institute and State University, the plasma ascorbic acid levels of pregnant women were compared for smokers versus nonsmokers. Thirty-two women in the
A study was conducted by the Department of Zoology at the Virginia Polytechnic Institute and State University to determine if there is a significant difference in the density of organisms at two
Five samples of a ferrous-type substance are to be used to determine if there is a difference between a laboratory chemical analysis and an X-ray fluorescence analysis of the iron content. Each
A taxi company manager is trying to decide whether the use of radial tires instead of regular belted tires improves fuel economy. Twelve cars were equipped with radial tires and driven over a
In Exercise 9.88 on page 315, use the t-distribution to test the hypothesis that the diet reduces a person's weight by 4.5 kilograms on the average against the alternative hypothesis that the mean
According to the published reports, practice under fatigued conditions distorts mechanisms which govern performance. An experiment was conducted using 15 college males who were trained to make a
In a study conducted by the Department of Human Nutrition and Foods at the Virginia Polytechnic Institute and State University the following data on the comparison of sorbic acid residuals in parts
How large a sample is required in Exercise 10.20 if the power of our test is to be 0.90 when the true mean is 5.20? Assume: that a = 0.24.
If the distribution of life spans in Exercise: 10.21 is approximately normal, how large a sample: is required in order that the probability of committing a type 11 error be 0.1 when the true mean is
How large a sample: is required in Exercise 10.22 if the power of our test is to be 0.95 when the true average height differs from 162.5 by 3.1 centimeters? Use: a = 0.02.
How large should the samples be in Exercise 10.31 if the power of our test is to be 0.95 when the true difference between thread types A and B is 8 kilograms?
How large a sample is required in Exercise 10.24 if the power of our test is to be 0.8 when the true mean meditation time exceeds the hypothesized value by 1.2σ? Use α = 0.05.
On testing Ho: p = 14, H1: μ ≠ 4, α = 0.05 level t-test is being considered. What sample size is necessary in order that the probability is 0.1 of falsely failing to reject Ho when the true
A study was conducted at the Department of Veterinary Medicine at Virginia Polytechnic Institute and State University to determine if the "strength" of a wound from surgical incision is affected by
Nine subjects were used in an experiment to determine if an atmosphere involving exposure to carbon monoxide has an impact on breathing capability. The data were collected by personnel in the Health
A marketing expert for a pasta-making company believes that 40% of pasta lovers prefer lasagna. If 9 out of 20 pasta lovers choose lasagna over other pastas, what can be concluded about the expert's
Suppose that, in the past, 40% of all adults favored capital punishment. Do we have reason to believe that the proportion of adults favoring capital punishment today has increased if, in a random
A coin is tossed 20 times, resulting in 5 heads. Is this sufficient evidence to reject the hypothesis that tile coin is balanced in favor of the alternative that heads occur less than 50% of the

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