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STAT 202 A student uses computer software to generate 10 random numbers

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STAT 202 A student uses computer software to generate 10 random numbers

Having problems with these last 12 homework questions. Any help would be appreciated.

A student uses computer software to generate 10 random numbers from a N(500, 100) distribution. From these ten numbers, she computes a 95% confidence interval for the mean using the formula

where  is the mean of the 10 random numbers. She then repeats this process (generating a new set of 10 random numbers from a N(500, 100) distribution each time) until she has produced 1000 such intervals. Which of the following will be true?

Approximately 97.5% of the intervals will contain the true mean because the probability that a standard Normal random variable is less than 1.96 is 0.975. She incorrectly used the formula for a 97.5% confidence interval.
Approximately 95% of the intervals will contain the value 100.
Approximately 95% of the intervals will contain the value 500.
2.      A student computed a 95% confidence interval for the mean, μ, of a population as (13, 20). Based on this interval,

the method gives correct results 95% of the time.
95% of the observations lie in the interval.
there is a 95% chance μ is in the interval.
3.      A researcher wants to know if the average time in jail for robbery has increased from what it was several years ago when the average sentence was 7 years. He obtains data on 400 more recent robberies and finds an average time served of 7.5 years. If we assume the standard deviation is 3 years, a 95% confidence interval for the average time served is

(7.11, 7.89)
(7.21, 7.79)
(7.25, 7.75)
4.      Suppose we are testing the null hypothesis H0: µ = 50 and the alternative Ha: µ ≠ 50 for a normal population with σ = 6. The 95% confidence interval for the mean is (51.3, 54.7). Then

the P-value for the test is less than 0.05.
the P-value for the test could be greater or less than 0.05. It can’t be determined without knowing the sample size.
the P-value for the test is greater than 0.05.
5.      A researcher wants to know if tougher sentencing laws have had the desired impact in terms of deterring crime. He plans to select a sample of states which have enacted a 3 strikes law and compare violent crime rates before the law was enacted with crime rates two years later. The correct set of hypotheses to test are

H0:  and Ha:
H0:  and Ha:
H0:  and Ha:
6.      A researcher wants to know if the average time in jail for robbery has increased from what it was several years ago when the average sentence was 7 years. He obtains data on 400 more recent robberies and finds an average time served of 7.5 years. If we assume the standard deviation is 3 years, what is the P-value of the test?

0.0008
0.0004
0.9996
7.      The teacher of a class of 40 high school seniors is curious whether the mean Math SAT score μ for the population of all 40 students in his class is greater than 500 or not. To investigate this, he decides to test the hypotheses
H0: μ = 500     Ha: μ > 500
at level α = 0.05. To do so, he computes the average Math SAT score of all the students in his class and constructs a 95% confidence interval for the population mean. The mean Math SAT score of all the students was 502 and, assuming the standard deviation of the scores is α = 100, he finds the 95% confidence interval is 502 ± 31. He may conclude

H0 cannot be rejected at level α = 0.05 because 500 is within confidence interval.
We can be certain that H0 is not true.
H0 cannot be rejected at level α = 0.05, but this must be determined by carrying out the hypothesis test rather than using the confidence interval.
8.      Suppose the average Math SAT score for all students taking the exam this year is 480 with standard deviation 100. Assume the distribution of scores is normal. The senator of a particular state notices that the mean score for students in his state who took the Math SAT is 500. His state recently adopted a new mathematics curriculum, and he wonders if the improved scores are evidence that the new curriculum has been successful. Since over 10,000 students in his state took the Math SAT, he can show that the P-value for testing whether the mean score in his state is more than the national average of 480 is less than 0.0001. We may correctly conclude that

although the results are statistically significant, they are not practically significant, since an increase of 20 points is fairly small.
there is strong statistical evidence that the new curriculum has improved Math SAT scores in his state.
these results are not good evidence that the new curriculum has improved Math SAT scores.
9.      Some researchers were asked to recheck the statistics on a paper they were writing about the impacts of a group of new science courses. They had performed 63 statistical tests on their data at α = 5% and had found four significant results. They were excited. Should they have been?

No. Too many tests were done.
Maybe. It depends on the actual results.
Yes. This could be important.
10.      Ginger root is used by many as a dietary supplement. A manufacturer of supplements produces capsules that are advertised to contain at least 500 mg of ground ginger root. A consumer advocacy group doubts this claim and tests the hypotheses
H0: μ = 500     Ha: μ < 500
based on measuring the amount of ginger root in a SRS of 100 capsules. Suppose the results of the test fail to reject H0 when, in fact, the alternative hypothesis is true. In this case, the consumer advocacy group will have

committed a Type I error.
committed a Type II error.
no power to detect a mean of 500.
11.      We perform a statistical test to examine if the mean amount filled of soft drinks has increased when cans are filled by a new machine compared to the old machine. Which of the following is a Type I error?

Conclude that the new machine increased the mean amount filled when in fact it does.
Conclude that the new machine did not increase the mean amount filled when in fact it does not.
Conclude that the new machine did not increase the mean amount filled when in fact it does.
Conclude that the new machine increased the mean amount filled when in fact it does not.
12.      We perform a statistical test to examine if the mean amount filled of soft drinks has increased when cans are filled by a new machine compared to the old machine. Which of the following is the power of the test?

the probability of concluding that the new machine did not increase the mean amount filled when in fact it does
the probability of concluding that the new machine did not increase the mean amount filled when in fact it does not
the probability of concluding that the new machine increased the mean amount filled when in fact it does not

the probability of concluding that the new machine increased the mean amount filled when in fact it does