A binary message m, where m is equal either to 0 or to 1, is sent over an information channel. Assume that if m = 0, the value s = −1.5 is sent, and if m = 1, the value s = 1.5 is sent. The value received is X, where X = s + E, and E ∼ N(0, 0.66). If X ≤ 0.5, then the receiver concludes that m = 0, and if X > 0.5, then the receiver concludes that m = 1.

If the true message is m = 0, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 1? Round the answer to four decimal places.

If the true message is m = 1, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 0?

A string consisting of 60 1s and 40 0s will be sent. A bit is chosen at random from this string. What is the probability that it will be received correctly?

Refer to part (c). A bit is chosen at random from the received string. Given that this bit is 1, what is the probability that the bit sent was 0?

Refer to part (c). A bit is chosen at random from the received string. Given that this bit is 0, what is the probability that the bit sent was 1?

Every day you flip a fair coin four times and if it is heads all four times, you give a dollar to charity. In a year with 365 days, what is your expected annual donation to charity and what is the variance?

Consider the following hypothesis test.

H₀: p = 0.30

H_a: p ≠ 0.30

A sample of 400 provided a sample proportion p = 0.275.

(a)

Compute the value of the test statistic. (Round your answer to two decimal places.) ______

(b) What is the p-value? (Round your answer to four decimal places.) p-value =_____

(c) At α = 0.05, what is your conclusion?

Reject H₀. There is insufficient evidence to conclude that p ≠ 0.30.

Reject H₀. There is sufficient evidence to conclude that p ≠ 0.30.     

Do not reject H₀. There is sufficient evidence to conclude that p ≠ 0.30.

Do not reject H₀. There is insufficient evidence to conclude that p ≠ 0.30.

(d)

What is the rejection rule using the critical value? (Round your answer to two decimal places. If the test is one-tailed, enter NONE for the unused tail.)

test statistic ≤ ______

test statistic ≥ ______

What is your conclusion?

Reject H₀. There is insufficient evidence to conclude that p ≠ 0.30.

Reject H₀. There is sufficient evidence to conclude that p ≠ 0.30.     

Do not reject H₀. There is sufficient evidence to conclude that p ≠ 0.30.

Do not reject H₀. There is insufficient evidence to conclude that p ≠ 0.30.

The time (in minutes) required for six‑year old children to assemble a certain toy is believed to be normally distributed with a known standard deviation of 3.0. The data in Table B gives the assembly times for a random sample of 25 children.

Table B:

Hi, I need help with questions 10, 11 and 12. Is there a way I can do them on excel? Thanks!

Consider the following gasoline sales time series. If needed, round your answers to two-decimal digits.

Is there a positive relationship between grit and GPA in high school seniors? A researcher examined this issue by having students beginning their senior year of high school complete a grit inventory using a Likert-based scale (range 1 – 7), where higher numbers indicate more “grit”. GPA was self-reported (scale 0 – 4.0). Enter the data shown here into SPSS to assess whether there is a positive relationship between grit and GPA.  

A savvy business owner wanted to assess whether the type of fragrance influenced the amount of money spent. He tried peppermint, lavender, male cologne, and a floral perfume in his four stores. Amount of money spent (in hundreds) is reported for each type of fragrance. Conduct a one-way repeated measures ANOVA to determine whether fragrance influences total amount of money spent.

The body temperatures of a group of healthy adults have a​ bell-shaped distribution with a mean of 98.08degreesF and a standard deviation of 0.63degreesF. Using the empirical​ rule, find each approximate percentage below. a. What is the approximate percentage of healthy adults with body temperatures within 1 standard deviation of the​ mean, or between 97.45degreesF and 98.71degrees​F? b. What is the approximate percentage of healthy adults with body temperatures between 96.19degreesF and 99.97degrees​F?

6. Research Question: Is there a higher percentage of Christians among CSU students than among USC students?

To test this I set up the hypothesis test

H0 : pCSU = pUSC Ha : pCSU > pUSC

This test gave a p-value of 0.023.

(a) Give your conclusion in context for this test using α = 0.05. (Your conclusion should be written so that a lay person could understand.)

(b) Give your conclusion in context for this test using α = 0.01. (Your conclusion should be written so that a lay person could understand.)

(c) Using the same sample data as in the hypothesis test, would a 95% confidence interval for pCSU − pUSC include the value 0? Explain.

Suppose someone you know and their spouse has been trying to start a family for a couple of years. The wife thinks she may be pregnant, so there will be a pregnancy test. Suppose the test has the hypotheses

H0 : Wife is pregnant.

Ha : Wife is not pregnant. (

(d) What are the consequences of making a Type I error?

(e) What are the consequences of making a Type II error?

The management of Hatman Toy Company is trying to determine the best production and overtime schedule for the coming month to attain maximum profit. The company makes two toy items. Each of the two toys needs to go through three departments (A: Casting, B: Painting, C: Costume) to be completed. The following table shows the amount of time (in minutes) each department needs to process each toy item: Product (minutes/unit) Department Toy 1 Toy 2 A 60 21 B 18 12 C 12 30 Toy item 1 is sold at $30 profit/unit and toy item 2 is sold at $15 profit/unit. Departments A, B, and C have a total of 100, 36, and 50 hours of (regular) labor time available each month for production. However, the company can schedule some overtime in each department at an additional cost. Each hour of overtime costs $18 in department A, $22.50 in department B, and $12 in department C. Up to a maximum of 10, 6, and 8 hours of overtime can be scheduled in departments A, B, and C, respectively. a) Write a mathematical formulation of the problem, which simultaneously determines how many units of each toy item to make, and how many hours of overtime should be used in each department (Hint: So we are trying to figure out 5 numbers in total). Clearly define your decision variables (full sentence!), and then write the objective function and constraints algebraically?

An article investigated the consumption of caffeine among women. A sample of 52 women were asked to monitor their caffeine intake over the course of one day. The mean amount of caffeine consumed in the sample of women was 226.553 mg with a standard deviation of 229.462 mg. In the article, researchers would like to include a 99% confidence interval.   

• The values below are t critical point corresponding to 90%, 95%, 98%, and 99% confidence. Which critical point corresponds to 99% confidence?
  • 2.676
  • 2.008
  • 1.675
  • 2.402
• We are 99% confident that the true mean amount of caffeine consumed by women is between ___ mg and ____ mg. (Round the limits of your interval to 4 decimal places.)
• If the level of confidence were changed to 98%, what effect would this have on the width of the calculated confidence interval?
  • there would be no effect on the interval
  • the interval would widen
  • the interval would narrow

The time required to assemble an electronic component is normally distributed with a mean and a standard deviation of 15 minutes and 8 minutes, respectively.

a) Find the probability that a randomly picked assembly takes between 12 and 19 minutes. (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

b) it is unusual for the assembly time to be above 28 minutes or below 5 minutes. What proportion of assembly times fall in these unusual categories? (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

In New York City, a study was conducted to evaluate whether any information that is available at the time of birth can be used to identify children with special educational needs. In a random sample of 45 third-graders enrolled in the special education program of the public school system, 4 have mothers who have had more than 12 years of schooling.

A) Construct a 90% CI for the population proportion of children with special educational needs whose mothers have had more than 12 years of schooling.

B) In 1980, 22% of all third-graders enrolled in the NYC public school system had mothers who had more than 12 years of schooling. Suppose you wish to know whether this proportion is the same for children in the special education program. What are the null and alternative hypotheses of the appropriate test?

C) Conduct the test at the 0.05 level of significance

Should use exact confidence interval and exact test in parts a and c. Please include your R commands for these parts.

Determine the uniformly most powerful test of H0:θ≤1 versus H1:θ >1 for a random sample of 25 from N(0,θ), at the significance level α=.05.

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $42,000 and a standard deviation of $5,700. We randomly survey ten teachers from that district. (Round your answers to the nearest dollar.)

(a) Find the 90^th percentile for an individual teacher's salary.
$ =

(b) Find the 90^th percentile for the average teacher's salary.
$ =