Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Identify the null and alternative​ hypotheses, test​ statistic, P-value, critical​ value(s), and state the final conclusion that addresses the original claim. A safety administration conducted crash tests of child booster seats for cars. Listed below are results from those​ tests, with the measurements given in hic​ (standard head injury condition​ units). The safety requirement is that the hic measurement should be less than 1000 hic. Use a 0.01 significance level to test the claim that the sample is from a population with a mean less than 1000 hic. Do the results suggest that all of the child booster seats meet the specified​ requirement? 690     739     1248     635     589     515

What are the​ hypotheses?

Identify the test statistic.

Identify the critical​ value(s).

State the final conclusion that addresses the original claim.

What do the results suggest about the child booster seats meeting the specified​ requirement?

1. Two cards are selected at random from a standard deck of 52 cards without replacement. What is the probability that both cards are diamonds? Submit your answer as a decimal rounded to the nearest thousandth.

1. A jar contains 14 blue candies, 10 green candies, and 5 yellow candies. Three candies are selected at random without replacement. What is the probability that the first is yellow, the second is blue, and the third is yellow? Submit your answer as a decimal rounded to the nearest thousandth.

1. In a group of 400 employees (142 men and 258 women), 29 of the men and 39 of the women work in accounting.  If an accountant is selected at random, what is the probability the accountant is a woman? Round your answer to the nearest thousandth.

1. In a group of 400 employees (142 men and 258 women), 29 of the men and 39 of the women work in accounting. If a male employee is selected at random, what is the probability that he is an accountant? Round your answer to the nearest thousandth.

The Toylot company makes an electric train with a motor that it claims will draw an average of only 0.8 ampere (A) under a normal load. A sample of nine motors was tested, and it was found that the mean current was x = 1.30 A, with a sample standard deviation of s = 0.42 A. Do the data indicate that the Toylot claim of 0.8 A is too low? (Use a 1% level of significance.)

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ = 0.8; H₁:  μ ≠ 0.8H₀: p = 0.8; H₁:  p ≠ 0.8     H₀: p = 0.8; H₁:  p > 0.8H₀: μ = 0.8; H₁:  μ > 0.8H₀: μ ≠ 0.8; H₁:  μ = 0.8H₀: p ≠ 0.8; H₁:  p = 0.8

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal, since we assume that x has a normal distribution with known σ.The Student's t, since we assume that x has a normal distribution with known σ.     The standard normal, since we assume that x has a normal distribution with unknown σ.The Student's t, since we assume that x has a normal distribution with unknown σ.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.2500.125 < P-value < 0.250     0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

1. The provost at the University of Chicago claimed that the entering class this year is larger than the entering class from previous years but their mean SAT score is lower than previous years. He took a sample of 20 of this year’s entering students and found that their mean SAT score is 1,501 with a standard deviation of 53. The University’s record indicates that the mean SAT score for entering students from previous years is 1,520. He wants to find out if his claim is supported by the evidence at a 5% level of significance. Round final answers to two decimal places. Solutions only.

(C) State the null hypothesis for this study.

(D) State the alternative hypothesis for this study.

(E) What critical value should the president use to determine the rejection region?

(H) The lowest level of significance at which the null hypothesis can still be rejected is ___.

1. A researcher is interested in testing two different dosages of a new sleeping medication in a phase II clinical trial against a control group receiving a placebo. Suppose the following table observes the hours of sleep reported the preceding evening for subjects randomly assigned to each of the three groups. α = .05

*Continue as though all assumptions for ANOVA are met.

A) Calculate the MSB and MSW for a one-way analysis of variance procedure.

B) Using the information above, calculate an F statistic, and provide an interpretation of your p-value

On the distant planet Cowabunga, the weights of cows have a normal distribution with a mean of 400 pounds and a standard deviation of 43 pounds. The cow transport truck holds 12 cows and can hold a maximum weight of 5076.

If 12 cows are randomly selected from the very large herd to go on the truck, what is the probability their total weight will be over the maximum allowed of 5076? (This is the same as asking what is the probability that their mean weight is over 423.)

The National Assessment of Educational Progress (NAEP) gave a test of basic arithmetic and the ability to apply it in everyday life to a sample of 840 men 21 to 25 years of age. Scores range from 0 to 500; for example, someone with a score of 325 can determine the price of a meal from a menu. The mean score for these 840 young men was x¯¯¯x¯ = 272. We want to estimate the mean score μμ in the population of all young men. Consider the NAEP sample as an SRS from a Normal population with standard deviation σσ = 60.

(a) If we take many samples, the sample mean x¯¯¯x¯ varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score μμ in the population. What is the standard deviation of this sampling distribution?
(b) According to the 95 part of the 68-95-99.7 rule, 95% of all values of x¯¯¯x¯ fall within _______ on either side of the unknown mean μμ. What is the missing number?
(c) What is the 95% confidence interval for the population mean score μμ based on this one sample? Note: Use the 68-95-99.7 rule to find the interval.

2.

The National Institute of Standards and Technology (NIST) supplies "standard materials" whose physical properties are supposed to be known. For example, you can buy from NIST a liquid whose electrical conductivity is supposed to be 5. (The units for conductivity are microsiemens per centimeter. Distilled water has conductivity 0.5.) Of course, no measurement is exactly correct. NIST knows the variability of its measurements very well, so it is quite realistic to assume that the population of all measurements of the same liquid has the Normal distribution with mean μμ equal to the true conductivity and standard deviation σσ = 0.2. Here are 6 measurements on the same standard liquid, which is supposed to have conductivity 5:

5.32   4.88   5.10   4.73   5.15   4.75

NIST wants to give the buyer of this liquid a 96% confidence interval for its true conductivity. What is this interval?

3.

Here are the IQ test scores of 31 seventh-grade girls in a Midwest school district:

These 31 girls are an SRS of all seventh-grade girls in the school district. Suppose that the standard deviation of IQ scores in this population is known to be σσ = 15. We expect the distribution of IQ scores to be close to Normal. Estimate the mean IQ score for all seventh-grade girls in the school district, using a 98% confidence interval.

3.

How heavy a load (pounds) is needed to pull apart pieces of Douglas fir 4 inches long and 1.5 inches square? Here are data from students doing a laboratory exercise:

Suppose that the strength of pieces of wood like these follows a Normal distribution with standard deviation 3000 pounds. Give a 99% confidence interval for the mean load required to pull the wood apart.

to  lb

A researcher wishes to estimate the proportion of adults who have​ high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.04 with 90​% confidence if ​

(a) she uses a previous estimate of 0.38​? ​n=

(b) she does not use any prior​ estimate? n=

Suppose that the probability function in the table below reflects the possible lifetimes (in months after emergence) for fruit flies. Let the random variable X measure fruit fly lifetimes (in months).

Fruit fly lifetimes (in months)

(a) What proportion of fruit flies die in their second month? P(x=2)

(b) What is the probability that a fruit fly lives more than four months?
(c) What is the mean lifetime for a fruit fly?
(d) What is the standard deviation of fruit fly lifetimes?

Test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the​ P-value method or the traditional method of testing hypotheses. Company A uses a new production method to manufacture aircraft altimeters. A simple random sample of new altimeters resulted in errors listed below. Use a 0.05 level of significance to test the claim that the new production method has errors with a standard deviation greater than 32.2​ ft, which was the standard deviation for the old production method. If it appears that the standard deviation is​ greater, does the new production method appear to be better or worse than the old​ method? Should the company take any​ action? negative 42​, 78​, -25​, -70​, -43​, 10​, 15​, 54​, -9​,-50​, -106​, -106

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of 70 professional actors, it was found that 42 were extroverts.

(a)

Let p represent the proportion of all actors who are extroverts. Find a point estimate for p. (Round your answer to four decimal places.)
1

(b)

Find a 95% confidence interval for p. (Round your answers to two decimal places.)
lower limit         2
upper limit         3

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 95% confident that the true proportion of actors who are extroverts falls outside this interval. We are 5% confident that the true proportion of actors who are extroverts falls within this interval.     We are 5% confident that the true proportion of actors who are extroverts falls above this interval. We are 95% confident that the true proportion of actors who are extroverts falls within this interval.

(c)

Do you think the conditions n·p > 5 and n·q > 5 are satisfied in this problem? Explain why this would be an important consideration.

Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial. No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.     No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal. Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.

TRACTOR SKIDDING HYPOTHESIS TESTING Forest engineers are interested in studying the skidding distances of tractors along a new road in a European Forest. The engineers collect data on a random sample of 12 tractors. The collected data (in meters) are: 350, 285, 574, 439, 295, 184, 261, 273, 400, 311, 141, 425. The average skidding distance for the sample of 12 tractors is 328.17 and the standard deviation is 118.46. Local loggers working on the road believe that the mean skidding distance of all tractors on this road is 325 meters. The engineers believe it is much greater than this. Conduct a hypothesis test based on the engineer’s belief. Conduct this hypothesis test at a significance level of .01. What conclusion can be made? Enter 1,2,3, or 4 in the box. Xbar is significantly greater than 325 feet. The average skidding distance of the tractors is significantly greater than 325 feet. µ is significantly greater than 325 feet. The average skidding distance of the tractors is significantly greater than 325 feet. Xbar is not significantly greater than 325 feet. The average skidding distance of the tractors is not significantly greater than 325 feet. µ is not significantly greater than 325 feet. The average skidding distance of the tractors is not significantly greater than 325 feet.

Consider an experiment in which a random family is selected among all families with exactly two children of which one is known to be a girl.

a. Write out the sample space and state the probability that the selected family has two girls.

b. Now consider an experiment in which we select a family randomly among all families with exactly two children, of which one is known to be a girl born on Tuesday. Write out the entire sample space taking into account the gender of the children and the day of the week they were born on.

c. What is the probability that the selected family has two girls?

d. Are the variables “day of the week” and “gender of child” dependent or independent?

A)While testing a building's fire alarms, the probability that any fire alarm will fail is 0.01. Suppose a building has 10 fire alarms, all which are independent of one another. The test will be passed if all fire alarms work.

a) 5 buildings are tested on the same day and each building has 10 fire alarms. How likely is it that 4 or more buildings pass the test? ( A building will pass if all 10 fire alarms are good)

B) Customers arrive at a restaurant according to a Poisson process at a rate of 30 customers per hour. There is a probability of 0.8 that a customer will dine in.

b) Say the customers are arriving independently of one another. What is the probability that 30 customers will arrive in a 1 hour time period AND all 30 will dine in?

C) Suppose a phd applicant is either accepted to a graduate program or not. if accepted the student can choose to attend or not attend. Suppose a graduate program has sent acceptance letters to 50 applicatns, but only had enough funding for 30 students. Let the students who were accepted to the program be independent of one another and the chance that a student will join the program be 0.6.

c) what is the probability that the graduate program will have enough funding for all students that joins the program.