Formulas
What setting on your calculator do you use for finding information about normal sampling distributions?
normalcdf
Draw a curve showing the probability of x being less than 14 with a mean of 16 and a standard deviation of 3.
Refer to slide 2
If Ruth’s test grade was in the 75th percentile, what was her grade when the standard deviation was 12 and the mean was 77?
85.09
In an exit poll, for which the sample size was 3190, suppose that 78% of all voters voted for a recall. Answer the following questions.
a. Define a binary random variable taking values 0 and 1 that represent the vote for a particular voter (1
=vote for recall). State its probability distribution. Choose the sentence below that best describes the probability distribution. (multiple choice!)
A.For each random sample of 3190 people, about 1755 people voted for recall and 436 did not.
B. The binary random variable is the vote for the recall with yes or no. For each observation, P(1)=0.22 and P(0)=0.78.
C. For each random sample of 3190 people, about 2488 people voted for recall and 702 did not.
D. The binary random variable is the vote for the recall with yes or no. For each observation, P(1)=0.78 and P(0)=0.22.
D
Draw the empirical rule curve and explain how it relates to the concept of Z-scores?
The empirical rule connects to z scores because it explains where a value falls on an empirical rule curve.
What is the formula for p hat?
x/n
The grades on an Ecology exam were approximately normal distributed with a mean of 74 and a standard deviation of 6.
Lizzie scored an 85. Find her z score and round to two decimal places.
1.83
If Jack’s test grade was in the 90th percentile, what was his grade if the mean was 83 and the standard deviation was 7.8?
About a 93, but the calculator shows a 92.996
According to a study conducted by an organization, the proportion of Americans who were afraid to fly in 2006 was 0.10. A random sample of 1,400 Americans results in 154 indicating that they are afraid to fly. Explain why this is not necessarily evidence that the proportion of Americans who are afraid to fly has increased.
Is this evidence or not?
What is the sample proportion?
a. No because the sample size is too small to represent the entire American population, meaning that the sampling error would be too large.
b. 0.11
The ______ ________ denoted as p hat, is given by the formula p hat = ______, where x is the number of individuals with a specified characteristic in a sample of n individuals.
1. Sample
2. Proportion
3. x/n
What is the formula for Z-Scores?
x-mean/(standard deviation)
Luke's exam score had a z-score of 2.32. With the class average being a 72 and the standard deviation having a value of 4.7, find Luke's exam score.
82.9
IQ scores have a normal distribution with a mean of 120 and a standard deviation of 18.
What percent of people have an IQ above 140?
What about if the scores are in between 82 and 130?
A. 0.13 or 13%
B. 0.69 or 69%
Consider a sampling distribution with p=0.14 and samples of size n each. Using the appropriate formulas, find the mean and the standard deviation of the sampling distribution of the sample proportion.
a. For a random sample of size n=7000.
b. For a random sample of size n=4000.
c. For a random sample of size n=300.
Mean for all three parts stays the same at 0.14
a. 0.0041
b. 0.0055
c. 0.0200
What effect does increasing the sample size have on the standard deviation? What explains this?
Increasing the sample size decreases the standard deviation. This is because of the central limit theorem
What is the sample distribution standard deviation formula?
(sigma) standard divided by the square root of the sample size.
I have a population that has an average height of 60 inches with a standard deviation of 5. What proportion of the population are 53 inches or shorter?
0.0808
According to a recent survey, the population distribution of number of years of education for self-employed individuals in a certain region has a mean of 12.3 and a standard deviation of 4.7.
a. Identify the random variable X whose distribution is described here.
b. Find the mean and the standard deviation of the sampling distribution of x for a random sample of size 150. Interpret them.
c. Repeat (b) for n = 500. Describe the effect of increasing n.
a. Years of education
b. mean = 12.3, SD = 0.39
c. mean = 12.3, SD = 0.21
n does not affect the mean, but can fluctuate the standard deviation
In an exit poll, suppose that the mean of the sampling distribution of the proportion of the 3390 people in the sample who voted for recall was 0.37 and the standard deviation was 0.0082. Answer the following questions.
a. Based on the approximate normality of the sampling distribution, give an interval of values within which the sample proportion will almost certainly fall.
b. Based on the result in (a), if you take an exit poll and observe a sample proportion of 0.31, would this be a rather unusual result? Why?
a. {0.3454, 0.3946}
b. 0.31 is unusual because it falls outside the interval of three standard deviations from the mean in both directions.
Define a random variable used in statistical terms.
What is the formula used for finding the standard deviation, given the proportion (p)?
Refer to slide one
What proportion of a population is greater than the value of 88 when the mean is 67 and the standard deviation is 10? Draw this on a curve with proper shading.
Refer to slide three for the curve.
0.018
A simple random sample of size n = 55 is obtained from a population with μ = 66 and σ = 11. Then, find the answers to A and B, adjusting to the sample size.
(a) Assuming the normal model can be used, determine P(x < 62.1).
(b) Assuming the normal model can be used, determine P(x ≥ 64.8).
a. 0.36 (population)
b. 0.54 (population)
a. 0.0042 (sample)
b. 0.7913 (sample)
An all you can eat restaurant charges $8.85 per customer and pricing is based off how much the customer eats and the labor behind it. The data has skewed right distribution with a mean of $8.40 and a SD of $3.
If 100 customers come in, find the mean and SD.
What is the probability that the restaurant will make profit that day, with the sample mean expense being less than $8.85 (Use CLT).
mean = 8.4
SD = 0.3
p = 0.933
Define sampling error. Why is it important?
This is when the population data is slightly different from the sample data. This is because sample data can only be so close to perfectly representing the population.