The population proportion is 0.28. What is the probability that a sample proportion will be within ±0.04 of the population proportion for each of the following sample sizes? (Round your answers to 4 decimal places.)

(a)n = 100

(b)n = 200

(c)n = 500

(d)n = 1,000

(R programming)

Generate 50 samples from a Poisson distribution with lambda to be 2 and define the log likelihood function

Use optimization to find the maximum likelihood estimator of lambda. Repeat for 100 times using forloop. You will need to save the results of the estimated values of lambda.

a)

b)

c)

In international Morse code, each letter in the alphabet is symbolized by a series of dots and dashes: the letter “a” for example is encoded as “×- ” while the most common letter “e” has the shortest code “×” (just a dot). What is the minimum number of dots and/or dashes needed to represent any letter in the English alphabet (26 letters)?

1. What is the mean of the sample values 2 cm, 2 cm, 3 cm, 5 cm, and 8 cm?

2. What is the median of the sample values listed in Exercise 1?

3. What is the mode of the sample values listed in Exercise 1?

4. If the standard deviation of a data set is 5.0 ft, what is the variance?

5. If a data set has a mean of 10.0 seconds and a standard deviation of 2.0 seconds, what is the z score corresponding to the time of 4.0 seconds?

6. Fill in the blank: The range, standard deviation, and variance are all measures of _____.

7. What is the symbol used to denote the standard deviation of a sample, and what is the symbol used to denote the standard deviation of a population?

8. What is the symbol used to denote the mean of a sample, and what is the symbol used to denote the mean of a population?

9. Fill in the blank: Approximately _____ percent of the values in a sample are greater than or equal to the 25th percentile.

10. True or false: For any data set, the median is always equal to the 50th percentile.

A consumer is trying to decide between two long-distance calling plans. The first one charges a flat rate of 10 cents per minute. The second charges a flat rate of 99 cents for calls up to 20 minutes in duration and then 10 cents for each additional minute exceeding 20. (Assume that calls lasting a non-integer number of minutes are charged proportionately to a whole-minutes charge). If the duration of a randomly selected call of this consumer is exponentially distributed and its expected value is 15 minutes, compute the expected value of the corresponding charge by each plan.

Hint: Denote by X the duration of a random call, by Y1 the charge of the first plan, and by Y2 the charge of the second plan. Then, express Y1 and Y2 as functions of X, i.e., Y1 = h1(X) and Y2 = h2(X)

The National Sleep Foundation used a survey to determine whether hours of sleeping per night are independent of age (Newsweek, January 19, 2004). The following show the hours of sleep on weeknights for a sample of individuals age 49 and younger and for a sample of individuals age 50 and older. Hours of Sleep Age Fewer than 6 6 to 6.9 7 to 7.9 8 or more Total 49 or younger 37 58 71 74 240 50 or older 34 60 79 87 260

a.Conduct a test of independence to determine whether the hours of sleep on weeknights are independent of age. Use = .05.

Compute the value of the 2 test statistic (to 2 decimals).

b.Using the total sample of 500, estimate the percentage of people who sleep less than 6, 6 to 6.9, 7 to 7.9, and 8 or more hours on weeknights (to 1 decimal).

Less than 6 hours %

6 to 6.9 hours %

7 to 7.9 hours %

8 or more hours %

A 0.01 significance level is used for a hypothesis test of the claim that when parents use a particular method of gender​ selection, the proportion of baby girls is different from 0.5. Assume that sample data consists of 66 girls in 144 births, so the sample statistic of 11/24 results in a z score that is 1 standard deviation below 0.

a. Identify the null hypothesis and the alternative hypothesis. Choose the correct answer below.

A.H0​:p=0.5

H1​:p>0.5

B. H0​: p≠0.5

   H1​: p=0.5

C.H0​: p=0.5

H1​: p<0.5

D.H0​: p=0.5

H1​: p≠0.5

b. What is the value of α​?

α=________

​(Type an integer or a​ decimal.)

c. What is the sampling distribution of the sample​ statistic?

Normal distribution

χ2

Student​ (t) distribution

d. Is the test​ two-tailed, left-tailed, or​ right-tailed?

___________

e. What is the value of the test​ statistic?

The test statistic is ____________

f. What is the​ P-value?

The​ P-value is _____________

g. What are the critical​ value(s)?

The critical​ value(s) is/are ________

h. What is the area of the critical​ region?

The area is ________

For a newsvendor product the probability distribution of demand X (in units) is as follows:

xi 0 1 2 3 4 5 6
pi 0.05 0.1 0.2 0.3 0.2 0.1 0.05

The newsvendor orders Q = 4 units.

a) Derive the probability distributions and the cumulative distribution functions of lost sales as well as leftover inventory.

b) Knowing that the expected total cost function is convex in the order quantity Q, demonstrate that Q = 4 gives the minimal expected total cost.

The size of the left upper chamber of the heart is one measure of cardiovascular health. When the upper left chamber is enlarged, the risk of heart problems is increased. A paper described a study in which the left atrial size was measured for a large number of children age 5 to 15 years. Based on this data, the authors concluded that for healthy children, left atrial diameter was approximately normally distributed with a mean of 26.7 mm and a standard deviation of 4.2 mm.

(a)

Approximately what proportion of healthy children have left atrial diameters less than 24 mm? (Round your answer to four decimal places.)

(b)

Approximately what proportion of healthy children have left atrial diameters greater than 32 mm? (Round your answer to four decimal places.)

(c)

Approximately what proportion of healthy children have left atrial diameters between 25 and 30 mm? (Round your answer to four decimal places.)

(d)

For healthy children, what is the value for which only about 20% have a larger left atrial diameter? (Round your answer to two decimal places.)

You may need to use the appropriate table in Appendix A to answer this question.

A population of 1,000 students spends an average of $10.50 a day on dinner. The standard deviation of the expenditure is $3. A simple random sample of 64 students is taken.

a. What are the expected value, standard deviation, and shape of the sampling distribution of the sample mean?

b. What is the probability that these 64 students will spend an average of more than $11 per person?

c. What is the probability that these 64 students will spend an average between $10 and $11 per person?

suppose a woman wants to estimate her exact day of ovulation for contraceptive purposes. A theory exists that at the time of ovulation the body temperature rises 0.5 to 1.0 degrees F thus, changes in body temperature can be used to goes the day of ovulation.

suppose that for this purpose a woman measures her body temperature on awakening on the first 10 days after menstruation and obtains the following data: 95.8, 96.5, 97.4, 97.4, 97.3, 96.0, 97.1, 97.3, 96.2, 97.3.

A. what is the best point estimate of her underlying basal body temperature (population mean)

b. how precise is this estimate (calculate the standard error of the estimate)?

c. compute a 95% confidence interval for the underlying mean basal body temperature using the data. assume that her underlying mean basal body temperature has a normal distribution

Desert Samaritan Hospital in Mesa, Arizona, keeps records of emergency room traffic. Those records reveal that the times between arriving patients have a mean of 8.7 minutes with a standard deviation of 8.7 minutes. Using only the values of these two parameters and your knowledge of the properties of the Normal distribution, give an argument why it is unreasonable to assume that the time between arrivals of buses is normally distributed (or even approximately so). (Hint: Consider the range of data of a Normal distribution.)

A baseball coach reviews the number of runs hit per game for the past several seasons. Since the team plays so many games, he selects a random sample of 10 games and records the number of runs scored in each game. The average number of runs scored is 7 with a standard deviation of 3.1 runs.

Compute the margin of error given a confidence level of 99%. (Use a table or technology. Round your answer to three decimal places.)

Working backwards, Part I. A 90% confidence interval for a population mean is (83, 89). The population distribution is approximately normal and the population standard deviation is unknown. This confidence interval is based on a simple random sample of 25 observations. Calculate the sample mean, the margin of error, and the sample standard deviation. Use the t distribution in any calculations. Round non-integer results to 2 decimal places.

Sample mean =

Margin of error =

Sample standard deviation =