Question
Asked Apr 23, 2019
37 views

11.13 Attached. Thank you for answering my question!

13. Let x, . . . . . X.25 be i.id. N(3, l ). Consider testing the null hypothesis Ho: μ 0
versus μ > 0 at significance level α-.05. Compare the power of the sign
test and the power of the test based on normal theory assuming that σ is known.
help_outline

Image Transcriptionclose

13. Let x, . . . . . X.25 be i.id. N(3, l ). Consider testing the null hypothesis Ho: μ 0 versus μ > 0 at significance level α-.05. Compare the power of the sign test and the power of the test based on normal theory assuming that σ is known.

fullscreen
check_circle

Expert Answer

Step 1

The hypothesis:

The hypothesis of interest is: H0: μ = 0 vs. HA: μ > 0, at level of significance α = 0.05.

The sample size is n = 25. The random sample is: X1, X2, …, X25.

Step 2

Power for normality assumption:

Under the assumption of normal theory, X1, X2, …, X25 is from a normal distribution with parameters: mean 0.3 and standard deviation 1. The known standard deviation, σ is 1.

Thus, the sample mean for the sample of size 25 will have a normal distribution with mean 0.3 and standard deviation 1/√25 = 1/5 = 0.2.

The power of the test is the probability that the sample mean exceeds the critical value, that is, lies in the rejection region, when the alternative hypothesis, HA is true. Using the given information, the value of μ considered for the alternative HA: μ > 0 is μ = 0.3.

Now, zα = z0.05 is the upper α-point or the point of the standard normal variable, the probability that a value lies above which is α = 0.05, in this case.

The critical value for the test using sample mean is: C = μ0 + zασ/√n, where μ0 is the hypothesized mean, here, μ0 = 0; σ/√n = 0.2; z0.05 = 1.645.

Thus, the power of the test is calculated as follows:

fullscreen
Step 3

Power for sign test:

The normal distribution being symmetric, mean = median. So, here, median is μ.

For the sign test, denote S as the count or the number of observations among X1, X2, …, X25, that are greater than the hypothesized median, μ = 0.

It is known that half of all observations lie above the median and half lie below the median. If the null hypothesis, H0 is true, then the probability that an observation is greater than the hypothesized value 0 is 1/2 = 0.5. Here, sample size, n = 25.

Thus, according to the rules of a binomial model, S must follow the binomial distribution with parameters: number of trials 25 and probability of success...

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Statistics

Advanced Topics in Statistics

Related Statistics Q&A

Find answers to questions asked by student like you
Show more Q&A
add
question_answer

Q: The government of a large city needs to determine whether the​ city's residents will support the bui...

A: Solution:Reasons for incorrect options:Opinions of other city residents who do not have children in ...

question_answer

Q: what is the class boundaries 16.5, 20.5, 24.5, 28.5, 32.5, 36.5, and 40.5.

A: Class boundaries are those numbers which are used to separate the classes.It is the midpoint of the ...

question_answer

Q: Find the standard deviation, s, of sample data summarized in the frequency distribution table given ...

A: Using the provided formula, the sample standard deviation of the provided frequency distribution is ...

question_answer

Q: The Nero Match Company sells matchboxes that are supposed to have an average of 40 matches per box, ...

A: It is given that the match boxes are supposed to have an average of 40 matches per box with σ = 7. A...

question_answer

Q: The number of unhealthy days based on the AQI ( Air Quality Index) for a random sample of metropolit...

A: Confidence interval:The (1-α) % confidence interval for population mean µ, when σ is unknown is give...

question_answer

Q: Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advan...

A: The provided information are:

question_answer

Q: Total blood volume (in ml) per body weight (in kg) is important in medical research. For healthy adu...

A: The provided information are:

question_answer

Q: A marketing manager for a cell phone company claims that more than 55% of children aged 8-12 have ce...

A: Null and alternative hypotheses:Here, to check whether more than 55% of children aged 8-12 have cell...

question_answer

Q: The point on a normal distribution (mean equals 6 and standard deviation equals 9) that is greater t...

A: The given information is that the scores distributed normally with mean 6 and standard deviation 9.F...