Question

 

1) Let X be a normally distributed random variable with a standard deviation equal 3; i.e. X∼N(μ, σ=3. Someone claims that μ=13. A random sample of 16 observations generate a sample mean of 14.56.

a) Does the sample mean provide evidence against the above claim at 5% significance level?

Complete the four steps of the test of hypothesis:

b) Assume that X was not normal; for example, X∼Unifμ-3, μ+3 which would have the same standard deviation. Use simulation to estimate the p-value for rejecting the above claim. Generate 1000 samples of sample means for samples of size 16 from the claimed distribution. Compute the proportion of them that fall as far or more away from the claimed mean (13) than does 14.56.

 

2) Let X be a normally distributed random variable with a standard deviation equal 3; i.e. X∼N(μ, σ=3). Someone claims that μ=13. You believe that the mean is lower and want to design a test that can reject this claim in 85% or more of times if the real mean is less than 12.5.

a) Find the sample size that satisfies your requirement at 1% significance level.

State the hypotheses and show details of your calculations.

b) Assume that X is not normal; for example, X∼Unif(μ-3, μ+3) which would have the same standard deviation. Use simulation to estimate the power of your test procedure found above in this scenario. Generate 1000 samples of sample means calculated for samples of size 16 from the distribution with alternative mean. Compute the proportion of them that will reject the null hypothesis in favor of the alternative.

 

 

 

Answer 1

Don't need to do the generations if can't!! At least the problems please!!

1) Let X be a normally distributed random variable with a standard deviation equal 3; i.e. X∼N(μ, σ=3. Someone claims that μ=13. A random sample of 16 observations generate a sample mean of 14.56.

a) Does the sample mean provide evidence against the above claim at 5% significance level?

Complete the four steps of the test of hypothesis:

Two tailed test

=2.08

Table value of z at 0.05 level = 1.96

Rejection Region: Reject Ho if z < -1.96 or z > 1.96

Calculated z = 2.08 in the rejection region

The null hypothesis is rejected.

There is enough evidence to reject the claim.

Z Test of Hypothesis for the Mean
Data
Null Hypothesis                       m=	13
Level of Significance	0.05
Population Standard Deviation	3
Sample Size	16
Sample Mean	14.56
Intermediate Calculations
Standard Error of the Mean	0.7500
Z Test Statistic	2.0800
Two-Tail Test
Lower Critical Value	-1.9600
Upper Critical Value	1.9600
p-Value	0.0375
Reject the null hypothesis

2) Let X be a normally distributed random variable with a standard deviation equal 3; i.e. X∼N(μ, σ=3). Someone claims that μ=13. You believe that the mean is lower and want to design a test that can reject this claim in 85% or more of times if the real mean is less than 12.5.

a) Find the sample size that satisfies your requirement at 1% significance level.

Lower tail test.

For 0.01 level, z = 2.576

d = 13-12.5=0.5

sd=3

power = 85%, for β =0.15, z = 1.036

= 469.67

The required sample size =470