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.

1(a) To find n and c, we need to use the binomial distribution and the probabilities of Type I error and of rejecting Ho when p=2/3.

The probability of a Type I error, denoted as 𝛼, is the probability of rejecting Ho when it is actually true. In this case, we want 𝛼 = 0.10.

The probability of rejecting Ho when p=2/3, denoted as 𝛽, is the probability of accepting H1 when it is actually true. In this case, we want 𝛽 = 0.95.

In order to find n and c, we can use a binomial probability distribution table or a statistical software. However, in this case, let's solve it using Excel.

We know that for a binomial distribution with parameters n and p, the probability 𝛼 of observing a value greater than or equal to c (rejecting Ho) can be calculated as:
𝛼 = 1 - P(Y < c-1)

Since p=1/2, we can calculate the probability 𝛼 in Excel using the BINOM.DIST function:
𝛼 = 1 - BINOM.DIST(c-1, n, 0.5, TRUE)

Similarly, the probability 𝛽 of observing a value greater than or equal to c when p=2/3 can be calculated as:
𝛽 = 1 - P(Y < c-1)

Since p=2/3, we can calculate the probability 𝛽 in Excel using the BINOM.DIST function:
𝛽 = 1 - BINOM.DIST(c-1, n, 2/3, TRUE)

To find n and c, we can use Excel's Solver add-in to optimize these probabilities. Set up the objective function as:
Objective function: |𝛽 - 0.95|
Subject to constraints:
- |𝛼 - 0.10| <= 0.001 (to ensure the probability of a Type I error is 0.10)
- n is an integer >= 1
- c is an integer >= 1

Solve this optimization problem using Excel Solver, and it will give you the values of n and c that satisfy the requirements.

(b) To find the probability of a Type II error, denoted as 𝛽̂, we need to know the alternative hypothesis H1. In this case, H1 is p > 0.40.

The probability of a Type II error is the probability of accepting the null hypothesis Ho when the alternative hypothesis H1 is true.

Using the given critical region C={y: y >= 13}, we can calculate 𝛽̂ as:
𝛽̂ = P(Y < 13) = BINOM.DIST(12, 25, 0.40, TRUE)

Calculate this probability using Excel or any statistical software to find the probability of a Type II error.

1(a) To find n and c such that the probability of a Type I error is 0.10 (α = 0.10), and P(reject Ho/p = 2/3) = 0.95:

We know that the probability of a Type I error, α, is given by:

α = P(reject Ho | Ho is true)

Since the Type I error probability is 0.10, we have:

α = 0.10

We also know that the probability of rejecting Ho when the true value of p is 2/3 is:

P(reject Ho/p = 2/3) = 0.95

From the binomial distribution, the rejection region is Y >= c, where Y ~ Binomial(n, p). In this case, p = 1/2.

To find n and c, we need to solve these two equations simultaneously.

First, we need to find c such that the probability of Y >= c, when p = 1/2, is 0.10 (α = 0.10). Let's call this value c1.

We can calculate c1 using the inverse binomial cumulative distribution function. In R programming language, this can be done as follows:

c1 <- qbinom(0.1, n, 1/2)

Next, we need to find n such that the probability of Y >= c1, when p = 2/3, is 0.95. Let's call this value n1.

We can calculate n1 using the inverse binomial cumulative distribution function again:

n1 <- qbinom(0.95, n, 2/3)

So the values of n and c that satisfy the given conditions are n = n1 and c = c1.

(b) The probability of a Type II error, β, is the probability of failing to reject Ho when H1 is true. In this case, H1: p > 1/2.

To calculate the probability of a Type II error, we need to know the true value of p under H1. However, this information is not provided in the question, so we cannot calculate the probability of a Type II error without knowing the true value of p.