QNT 275 Statistics Assignment

Question 1:

Find the area under the standard normal curve

1. between z=0 and z=1.95

2. between z=0 and z=−2.05

3. between z=1.15 and z=2.37

4. from z=−1.53 to z=−2.88

5. from z=−1.67 to z=2.24

Compute the area under the standard normal curve. Round your answers to four decimal places.

(a) The area to the right of z = 1.85 equals

(b) The area to the left of z = − 1.2 equals

(c) The area between z = 0 and z = 3.19 equals

Question 2:

Obtain the following probabilities for the standard normal distribution.

P(−2.47≤z≤1.29)

Obtain the following probability for the standard normal distribution.

Round your answer to four decimal places.

Question 3:

Let x be a continuous random variable that has a normal distribution with a mean of 117.6 and a standard deviation of 14.6. Find the probability that x assumes a value

1. between 77.9 and 98.3

2. between 85.3 and 142.6

Compute probabilities.

Recall the following definitions from section 6.4 of the text.

The area under the normal curve from x = a to x = b with given mean and standard deviation is the probability that x assumes a value between x = a and x = b. If we are using Table IV in Appendix C, we need to standardize the random variable x using the formula z = (x − µ)/σ, before using the table.

Alternatively, you may use a graphing calculator to obtain more accurate calculations without standardizing the random variable x. For example, using a TI83 plus we calculate the area under the normal curve from x = a to x = b by using the

3. normalcdf(a,b,µ,σ)

where µ is the mean and σ is the standard deviation of the normal distribution. We use 1E99 for ∞ and −1E99 for −∞, if needed.

Let x be a continuous random variable that is normally distributed with a mean of and a standard deviation of .

Round your answers to two decimal places.

(a) Standardize the variable value x = .

z =

(b) Standardize the variable value x = .

z =

Question 4:

Let x be a continuous random variable that is normally distributed with a mean of 24 and a standard deviation of 6. Find the probability that x assumes a value between 28.0 and 54.0.

Use Table IV in Appendix C to compute the probabilities.

Round your answer to four decimal places.

The probability =

Question 5:

Find the mean and the sampling/nonsampling error.

Consider the following population of 10 numbers.

Round answers to two decimal places.

1. Find the population mean.

2. Rich selected one sample of nine numbers from this population. The sample included the numbers 20, 25, 13, 9, 15, 11, 7, 17, and 30. Calculate the sample mean and sampling error for this sample.

3. Refer to part b. When Rich calculated the sample mean, he mistakenly used the numbers 20, 25, 13, 9, 15, 11, 17, 17, and 30 to calculate the sample mean. Find the sampling and nonsampling errors in this case.

4. List all samples of nine numbers (without replacement) that can be selected from this population. Calculate the sample mean and sampling error for each of these samples.

Question 6:

The following data give the ages (in years) of all six members of a family.

List all possible samples of size five (without replacement) that can be selected from this population.

Calculate the mean for each of these samples. Enter the exact answers.

For a sample , , , , , mean .

For a sample , , , , , mean .

For a sample , , , , mean .

For a sample , , , , mean .

For a sample , , mean .

For a sample , , , , mean .

Write the sampling distribution of   Find   Round your answer to three decimal places.

List all the possible samples of size four (without replacement) that can be selected from this population. Calculate the mean for each of these samples. Write the sampling distribution of x¯

Question 7:

A population of N = 100000 has a standard deviation of σ = 40. A sample of size n was chosen from this population. In each of the following two cases, decide which formula would you use to calculate   and calculate   Recall the following from section 7.3 of the text. The standard deviation   of the sampling distribution of   mean of the sample is given by the formula   where n is the sampling size and σ is the population standard deviation provided the sample size is small in comparison to the population size N . Sample size is considered to be small compared to the population size, if n ≤ 0.05N (sample size is less than or equal to 5% of the population). If this condition is not satisfied, we use the following formula to calculate     The factor   in the formula is called the finite population correction factor.

A sample of size n = 3000 is chosen from the population of size N = 100000 that has a standard deviation of σ = 40. (a) Does the following condition hold: n ≤ 0.05N?

(b) Which formula would you use to calculate  

Choose the correct formula number from the table above:

(c) Using the appropriate formula, calculate   Round the answer to four decimal places.  

·

A population of N=5000 has σ=25. In each of the following cases, which formula will you use to calculate σx¯ and why? Using the appropriate formula, calculate σx¯ for each of these cases.

1. n = 300

2. n = 100

Question 8:

A population of N = 100000 has a standard deviation of σ = 60. A sample of size n was chosen from this population. In each of the following two cases, decide which formula would you use to calculate   and calculate  

Round the answers to four decimal places.

(a) n = 2500.

(b) n = 6500

Question 9:

A population has a normal distribution. A sample of size n is selected from this population. Describe the shape of the sampling distribution of the sample mean for each of the following cases.

1. n=94

2. n=11

A population has a normal distribution. A sample of size is selected from this population. Describe the shape of the sampling distribution of the sample mean for the following case.

Question 10:

The amounts of electricity bills for all households in a particular city have an approximate normal distribution with a mean of $140 and a standard deviation of $.30 Let x¯ be the mean amount of electricity bills for a random sample of 25 households selected from this city. Find the mean and standard deviation of x¯, and comment on the shape of its sampling distribution.

The amounts of electricity bills for all households in a particular city have an approximately normal distribution with a mean of and a standard deviation of . Let be the mean amount of electricity bills for a random sample of households selected from this city. Find the mean and standard deviation of .

Round your answers to the nearest integer, if required. Comment on the shape of the sampling distribution of . The sampling distribution of is

.