Exam One
Is this considered continuous or discrete?
Weight, height, hair length.
Continuous
Explain the different rules for probability (two for multiplication, two for addition). Indicate which ones are mutually exclusive.
Refer to slide 3
Mary made a 55 on and exam with the class average being a 65 and a standard deviation of 4.8. Find the Z-score.
-2.08
Define confidence interval and explain how to find it.
How confident we are that data falls within a certain range. By multiplying the margin of error by the standard error.
Is the following considered to be numerical or categorical? Is it continuous or discrete?
Eye color, hair color, preference of cats or dogs.
Not continuous or discrete because these are categorical.
26, 37, 49, 64, 68, 69, 72, 74, 75, 77, 78, 80, 82, 83, 85, 85, 86, 91, 94, 100
Find the mean, median, mode, and standard deviation. Draw a stem and leaf plot
Mean = 73.16
Median = 77
Mode = 85
Standard Deviation = 18.7
Write out the rule for conditional probability, and explain it as if someone had never heard of statistics or probability before.
P(A|B) = P(A and B)/P(B)
Probability of A given that B already happened.
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
A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x, is found to be 107, and the sample standard deviation, s, is found to be 10.
(a) Construct a 95% confidence interval about μ if the sample size, n, is 12.
(b) Construct a 95% confidence interval about μ if the sample size, n, is 24.
100.6 to 113.4
102.8 to 111.2
Give the difference between independent and dependent events. Provide an example.
Dependent events are when the previous outcomes effect the following outcomes, and independent events are the opposite. An example would be drawing a deck of cards. If a card is drawn and is not replaced, it is a dependent event. If a card is drawn and is replaced, then it is an independent event.
35, 32, 31, 34, 35, 35, 35, 35, 32, 31, 34, 35, 35, 35
Find the mean, median, mode, and standard deviation. Are there any outliers? If so, show how you found them.
Mean = 33.86, median = 35, mode = 35, Standard deviation = 1.61
No outliers
A bag contains 6 green marbles, 5 blue, and 9 red. You draw twice with replacement. What is the probability that you will draw a green marble and then a blue marble?
P(Green and Blue): 3/40
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
Determine the point estimate of the population proportion, the margin of error for the following confidence interval, and the number of individuals in the sample with the specified characteristic, x, for the sample size provided.
Lower bound = 0.391, upper bound = 0.909, n = 1200
The point estimate of the population proportion is
The Margin of error is
The number of individuals in the sample with the specified characteristic is
0.650
0.259
780
Define Mutually Exclusive and Provide an example. Define non mutually exclusive and provide an example.
When two events cannot occur at the same time. For example, when you flip a coin, you cannot get both heads and tails.
The events can happen at the same time. An example would be when you roll a dice and want to get an odd number. There are multiple odd numbers on a dice.
Another class took the same midterm exam. The scores are as follows:
46, 46, 74, 80, 89, 90, 92, 94, 96, 98, 99, 100
Find the Median, IQR, and Outliers. Show your work. Draw a box and whisker plot.
Median - 92
Outliers - 46
IQR = 20
Refer to slide 2
a. 422/427302
b. 164872/591752
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
A survey asked whether respondents favored or opposed the death penalty for people convicted of murder. Software shows the results below, where X refers to the number of the respondents who were in favor. Construct the 95% confidence interval for the proportion of the adults who were opposed to the death penalty from the confidence interval stated below for the proportion in favor.
Find the confidence interval for those opposed.
0.294
0.330
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.
Use the frequency table on slide four.
Find the mean
8.1
What is the probability that you will land in the safe zone?
Refer to slide one
7/12
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
In a lottery, you bet on a six-digit number between 000000 and 111111. For a $1 bet, you win $800,000 if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are μ = 0.8 (that is, 80 cents) and σ = 800.00. Joan figures that if she plays enough times every day, eventually she will strike it rich, by the law of large numbers. Over the course of several years, she plays 1 million times. Let x denote her average winnings.
a. Find the mean and standard deviation of the sampling distribution of x.
b. About how likely is it that Joan's average winnings exceed $1, the amount she paid to play each time? Use the central limit theorem to find an approximate answer.
0.8
0.8
0.401
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.