Suppose a single measurement is taken from a distribution with pdf f(x)=?e−?x, x>0. The hypotheses are...
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Suppose a single measurement is taken from a distribution with pdf f(x)=?e−?x, x>0. The hypotheses are H0∶?=1 versus HA∶?<1, and the null hypothesis is rejected if x ≥ 3.2.
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a) Calculate the probability of committing a type I error.
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b) Calculate the probability of committing a type II error if, in fact, ? = 1∕5.
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Suppose a single measurement is taken from a distribution with
pdf f(x)=?e−?x, x>0. The hypotheses are...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x >...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x > 0, and mean θ. A single observation of an exponential random variable X is used to test H0 : θ-2 against H1 : θ-5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. (a) What is the probability of committing a Type I error? (b) What is the probability of committing a Type...
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A...
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A hypothesis test is as follows: H0: θ = 2 H1: θ ≠ 2 It has been decided to reject H0 if the observed value of x is x ≤ 0.1 or x ≥ 1.9. Part a: Find the probability of committing a Type I error. Part b: Suppose the true value of θ is 3. Find the probability of committing a Type II error....
5. A bowl contains ten marbles of which 0 are white while the others are black....
5. A bowl contains ten marbles of which 0 are white while the others are black. In order to test the null hypothesis H0 : θ-1 against H1: θ > 1, five of the marbles are randomly drawn with replacement and the null hypothesis is rejected if at least two of the five drawn are white. (a) What is the probability of committing a Type I error? (21 (b) What is the probability of committing a Type II error when...
Let X1, X2 ,, X8 be a random sample of size 8 from a POI(λ) distribution....
Let X1, X2 ,, X8 be a random sample of size 8 from a POI(λ) distribution. The null hypothesis that λ = 0.25 will be rejected in favor of the alternative hypothesis that λ>0.25 if ∑X ≥5. i a. Find the probability of committing a type I error. (Hint: If Xi ~ POI(λ), then ∑Xi ~ POI(nλ)). b. Find the probability of committing a type II error if λ = 0.5 . c. Find the power of the test if...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
Problem H5 Let X be a single observation (n-1) from the following distribution: f(rle)-o elsewhere NOTE:XBeta(0,...
Problem H5 Let X be a single observation (n-1) from the following distribution: f(rle)-o elsewhere NOTE:XBeta(0, 1) The following two hypotheses are being tested: 110 : e-2 vs Ha : ?-1. (a) Draw a graph of f(z | ?) when (i) Ho is true and when (ii) H. is true. Put both graphs on the same plot. Explain why a rejection region of the form (X<k) makes intuitive sense (b) Find k, so that the test has level a 0.05....
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a sample of size 3 f...
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a sample of size 3 from this pdf. Use the 4,0 y 5 versus HA : θ > 5. (a) What is the decision rule when a 0.05. (b) Suppose θ-7, what is the Type II error for the test in part (a).
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a...
Example 8.4 An automobile model is known to sustain no visible damage 25% of the time...
Example 8.4 An automobile model is known to sustain no visible damage 25% of the time in 10-mph crash tests. A modified bumper design has been proposed in an effort to increase this percentage. Let p denote the proportion of all 10-mph crashes with this new bumper that result in no visible damage. The hypotheses to be tested are Ho: P = 0.25 (no improvement) versus Ha: p ? ? 0.25. The test will be based on an experiment involving...
According to a food website, the mean consumption of popcorn annually by Americans is 57 quarts....
According to a food website, the mean consumption of popcorn annually by Americans is 57 quarts. The marketing division of the food website unleashes an aggressive campaign designed to get Americans to consume even more popcorn. Complete parts (a) through (c) below. (a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the marketing campaign. H0: ▼ p σ μ ▼ ≠ = _______? H1: ▼ σ p μ ▼ ≠ > < _______?...
Suppose that you are testing the following hypotheses: Ho: = 10 and 11: > 10. If...
Suppose that you are testing the following hypotheses: Ho: = 10 and 11: > 10. If the null hypothesis is rejected at the 1% level of significance, what statement can you make about the confidence interval on the mean? a) The lower bound of a 95% one-sided confidence interval on the mean exceeds 10. b) 9 sus 13 c) No statement can be made. d) The lower bound of a 95% one-sided confidence interval on the mean exceeds zero. If...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x > 0, and mean θ. A single observation of an exponential random variable X is used to test H0 : θ-2 against H1 : θ-5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. (a) What is the probability of committing a Type I error? (b) What is the probability of committing a Type...
5. A bowl contains ten marbles of which 0 are white while the others are black. In order to test the null hypothesis H0 : θ-1 against H1: θ > 1, five of the marbles are randomly drawn with replacement and the null hypothesis is rejected if at least two of the five drawn are white. (a) What is the probability of committing a Type I error? (21 (b) What is the probability of committing a Type II error when...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
Problem H5 Let X be a single observation (n-1) from the following distribution: f(rle)-o elsewhere NOTE:XBeta(0, 1) The following two hypotheses are being tested: 110 : e-2 vs Ha : ?-1. (a) Draw a graph of f(z | ?) when (i) Ho is true and when (ii) H. is true. Put both graphs on the same plot. Explain why a rejection region of the form (X<k) makes intuitive sense (b) Find k, so that the test has level a 0.05....
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a sample of size 3 from this pdf. Use the 4,0 y 5 versus HA : θ > 5. (a) What is the decision rule when a 0.05. (b) Suppose θ-7, what is the Type II error for the test in part (a).
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a...
Example 8.4 An automobile model is known to sustain no visible damage 25% of the time in 10-mph crash tests. A modified bumper design has been proposed in an effort to increase this percentage. Let p denote the proportion of all 10-mph crashes with this new bumper that result in no visible damage. The hypotheses to be tested are Ho: P = 0.25 (no improvement) versus Ha: p ? ? 0.25. The test will be based on an experiment involving...
Suppose that you are testing the following hypotheses: Ho: = 10 and 11: > 10. If the null hypothesis is rejected at the 1% level of significance, what statement can you make about the confidence interval on the mean? a) The lower bound of a 95% one-sided confidence interval on the mean exceeds 10. b) 9 sus 13 c) No statement can be made. d) The lower bound of a 95% one-sided confidence interval on the mean exceeds zero. If...
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