A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x, is found to be 50, and the sample standard deviation, s, is found to be 8.

a) Describe the sampling distribution of the sample mean x.

b) Construct a 98% confidence interval for μ if the sample size, n, is 20.

c) Construct a 98% confidence interval for μ if the sample size, n, is 15. How does decreasing the sample size affect the margin of error, E?

d) Construct a 95% confidence interval for μ if the sample size, n, is 20. How does decreasing the level of confidence affect the margin of error, E?

e)If the population had not been normally distributed could we have computed the confidence intervals in parts (b) – (d)?

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual. For a sample of n= 69​, find the probability of a sample mean being less than 23.3 if m= 23 and sigma =1.17.

The population mean and standard deviation are given below. Find the indicated probability and determine whether a sample mean in the given range below would be considered unusual. If​ convenient, use technology to find the probability.

For a sample of n =31, find the probability of a sample mean being less than 12,750 or greater than 12753 when μ =12,750 and σ =2.3

The probability that a randomly selected 3​-year-old male garter snake will live to be 4 years old is 0.99244.
​(a) What is the probability that two randomly selected 3​-year-old male garter snakes will live to be 4 years​ old?
​(b) What is the probability that eight randomly selected 3​-year-old male garter snakes will live to be 4 years​ old?
​(c) What is the probability that at least one of eight randomly selected 3​-year-old male garter snakes will not live to be 4 years​ old? Would it be unusual if at least one of eight randomly selected 3​-year-old male garter snakes did not live to be 4 years​ old?

These are stressful days in general. Final exams coming up, a lot of studying to do, all the news around us, …….. A large university conducted a study of their undergraduate students to find out how stressful they feel and recorded their gender as well below is the table that shows how each of the gender felt when asked about if they felt stressed (5 points) Felt Stressed Gender YES NO Male 244 495 Female 282 480 a. Find out if there is a difference between males and females with regards to feeling stressed out. Use the 0.05 significance level for testing b. what is the p value and interpret its meaning in the context of the problem above

6)

One thousand people compete in an oyster-eating contest. Their scores (number of oysters eaten) are normally distributed with a mean of 51 and standard deviation of 11. What is the probability that a given individual will eat less than 29 or more than 72?

Select one:

( ) a. 98%

( ) b. 5%

( ) c. 13%

( ) d. 2%

( ) e. 47%

A recent study focused on the number of times men and women who live alone buy take-out dinner in a month. Assume that the distributions follow the normal probability distribution and the population standard deviations are equal. The information is summarized below.

At the 0.01 significance level, is there a difference in the mean number of times men and women order take-out dinners in a month?

1. State the decision rule for 0.01 significance level: H₀: μ_Men= μ_Women   H₁: μ_Men ≠ μ_Women. (Negative values should be indicated by a minus sign. Round your answers to 3 decimal places.)

2. Compute the value of the test statistic. (Round your answer to 3 decimal places.)

3. What is your decision regarding the null hypothesis?

4. What is the p-value? (Round your answer to 3 decimal places.)

Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma.† Over a period of months, an adult male patient has taken nine blood tests for uric acid. The mean concentration was x = 5.35 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with σ = 1.85 mg/dl.

(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient's blood. What is the margin of error? (Round your answers to two decimal places.)

(b) What conditions are necessary for your calculations? (Select all that apply.)

σ is unknownnormal distribution of uric aciduniform distribution of uric acidn is largeσ is known

(c) Interpret your results in the context of this problem.

We are 5% confident that the true uric acid level for this patient falls within this interval.The probability that this interval contains the true average uric acid level for this patient is 0.95.     We are 95% confident that the true uric acid level for this patient falls within this interval.The probability that this interval contains the true average uric acid level for this patient is 0.05.

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.12 for the mean concentration of uric acid in this patient's blood. (Round your answer up to the nearest whole number.)
blood tests

Sheila's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Sheila's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ = 125 mg/dl and σ = 10 mg/dl. What is the level L such that there is probability only 0.05 that the mean glucose level of 3 test results falls above L for Sheila's glucose level distribution? (Round your answer to one decimal place.)

The null and alternate hypotheses are:

H₀ : μ_d ≤ 0

H₁ : μ_d > 0

The following sample information shows the number of defective units produced on the day shift and the afternoon shift for a sample of four days last month.

At the 0.100 significance level, can we conclude there are more defects produced on the day shift? Hint: For the calculations, assume the day shift as the first sample.

1. State the decision rule. (Round your answer to 2 decimal places.)

2. Compute the value of the test statistic. (Round your answer to 3 decimal places.)

3. What is the p-value?

• Between 0.10 and 0.15

• Between 0.001 and 0.005

• Between 0.005 and 0.01

4. What is your decision regarding H₀?

• Do not reject H₀

• Reject H₀

Patsy Pennypincher is very concerned with the rising cost of education. She collected a sample of 20 institutions (n=20) and found a sample mean cost of $15,500 (mean = 15,500) and a sample standard deviation to be $3000 (s = 3000). Provide Patsy with a 98% confidence interval for this year’s tuition cost.

Please show your work.

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 18 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.20 gram.

(a)

Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limitupper limitmargin of error

(b)

What conditions are necessary for your calculations? (Select all that apply.)

normal distribution of weightsn is largeσ is unknownuniform distribution of weightsσ is known

(c)

Interpret your results in the context of this problem.

We are 20% confident that the true average weight of Allen's hummingbirds falls within this interval.The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.     The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval.

(d)

Find the sample size necessary for an 80% confidence level with a maximal margin of error  E = 0.09 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

hummingbirds

The heights of UNC sophomores are approximately normally distributed. The heights in inches of 8 randomly selected sophomores are shown below. Use these heights to find a 95% confidence interval for the average height μ of UNC sophomores. Give the endpoints of your interval to one decimal place.

72, 69, 70, 68, 70, 66, 75, 64

a) Use these heights to find a 95% confidence interval for the average height μ of UNC sophomores. Give the endpoints of your interval to one decimal place.

95% confidence interval: (  ,  )

b) Using the 8 data points above, conduct a test of following hypotheses at level of significance α=0.05:

H₀: μ = 67; H_a: μ > 67

Find the sample mean x, the sample standard deviation s, the appropriate t-statistic and p-value. Use Excel when necessary and give all answers to four decimal places.

Sample mean =  
Sample standard deviation =  
t =  

p=

Suppose an audit process is taking place at a small airport, and the head
auditor arrives randomly throughout the day and assesses the operation of the airport.
She shows up in one-hr intervals at a time. The time between aircraft arrivals at this
small airport is exponentially distributed with a mean of 75 minutes. You may assume
the operation of the small airport is independent of the time interval under consideration.

What is the probability that at least four aircrafts arrive within an hour?

What is the probability that more than three and less than six aircrafts arrive within an
hour?

1. Using the sample 3, 4, 6, 1, 10, 6

• Find the median, mean, mode, and range

• Find the standard deviation

2. Using the sample 8, 5, 1, 5, 2, 3

• Find the median, mean, mode, and range

• Find the standard deviation