1(a)Let Y have a binomial distribution with parameters n and p. We reject Ho: p=1/2 and accept H1: p>1/2 if Y>=c. Find n and c such that the probability of a Type I error is .10, and P(reject Ho/p=2/3)=.95.

(b)What is the probability of a Type II error?

2.Let xbar be the observed mean of a random sample of size n from a distribution having mean mu and variance sigma squared. Find n so that xbar-sigma/4 to xbar+sigma/4 is an approximate 95% confidence interval for mu.

3. Let p denote the probability that, for a particular tennis player, the first serve is good. Since p=0.40 this player decided to take lessons in order to increase p. When the lessons are completed, the hypothesis Ho:p=0.40 will be tested against H1:p>0.40 based on n=25 trials. Let y equal the number of first serves that are good, and let the critical region be defined by C={y:y>=13}. Determine the significance level.

4. Let X be N(mu,sigma squared)so that P(X<89)=0.90 and P(X<94)=0.95. Find mu and sigma squared.

5. Let X1,...,X9 be a random sample of size 9 from a distribution that is N(mu,sigma squared). If sigma is known, find the length of a 95% confidence interval for mu if this interval is based on the random variable Squarert9(xbar-mu)/sigma.

6. Half pint (8oz) milk cartons are filled at a dairy by a filling machine. To provide a check on the machine, a sample of 10 cartons is periodically measured. If the sample mean deviates by more than a certain amount d from the nominal value 8oz, i.e, if [xbar-8]>d, then the machine setting is adjusted. The chance of a false alarm indicating an unnecessary adjustment is to be limited to 1%. Find a formula for d.

7. The lifetime of lightbulbs of a particular type are normally distributed with a mean of 293 hours and a standard deviation of 6 hours. Find the 1st quartile.

(b)What is the probability of a Type II error?

2.Let xbar be the observed mean of a random sample of size n from a distribution having mean mu and variance sigma squared. Find n so that xbar-sigma/4 to xbar+sigma/4 is an approximate 95% confidence interval for mu.

3. Let p denote the probability that, for a particular tennis player, the first serve is good. Since p=0.40 this player decided to take lessons in order to increase p. When the lessons are completed, the hypothesis Ho:p=0.40 will be tested against H1:p>0.40 based on n=25 trials. Let y equal the number of first serves that are good, and let the critical region be defined by C={y:y>=13}. Determine the significance level.

4. Let X be N(mu,sigma squared)so that P(X<89)=0.90 and P(X<94)=0.95. Find mu and sigma squared.

5. Let X1,...,X9 be a random sample of size 9 from a distribution that is N(mu,sigma squared). If sigma is known, find the length of a 95% confidence interval for mu if this interval is based on the random variable Squarert9(xbar-mu)/sigma.

6. Half pint (8oz) milk cartons are filled at a dairy by a filling machine. To provide a check on the machine, a sample of 10 cartons is periodically measured. If the sample mean deviates by more than a certain amount d from the nominal value 8oz, i.e, if [xbar-8]>d, then the machine setting is adjusted. The chance of a false alarm indicating an unnecessary adjustment is to be limited to 1%. Find a formula for d.

7. The lifetime of lightbulbs of a particular type are normally distributed with a mean of 293 hours and a standard deviation of 6 hours. Find the 1st quartile.