Biol 528 Exam 1
Given a binomial distribution with a probability of success = 0.6, what is the standard deviation after 10 trials? 0.6 * 10 10 * 0.6 * 0.4 (10 * 0.6 * 0.4)^0.5 A or B
(10 * 0.6 * 0.4)^0.5
# of Antelope Bands Cum. Freq Cumulative Prob 1 2 0.0769 2 3 0.1154 3 5 0.1923 4 10 0.3846 5 11 0.4231 6 14 0.5385 7 15 0.5769 8 17 0.6538 9 18 0.6923 The table above is a cumulative frequency of antelope bands observed. What is the probability of observing 7 or more antelope bands? Answer your question using a decimal with a zero: example: 0.XX
0.4615 Yes! think about the cumulative frequency you did on the HW 2. (1-cumulative probability of 6 bands)
# of Antelope Bands Cum. Freq Cumulative Prob 1 2 0.0769 2 3 0.1154 3 5 0.1923 4 10 0.3846 5 11 0.4231 6 14 0.5385 7 15 0.5769 8 17 0.6538 9 18 0.6923 The table above is a cumulative frequency of antelope bands observed. What is the probability of observing 7 or more antelope bands OR 2 or less antelope bands? Answer your question using a decimal with a zero, example: 0.XX
0.5769 Yes! think about the cumulative frequency you did on the HW 2- cumulative probability of 6 bands + cumulative frequency of 2 or less bands
Binomial Properties
1.The event, or trial, occurs a specified number of times (denoted as k) 2.For each event there are two mutually exclusive outcomes (success, p) (failure, q) p+q = 1 q=1-p 3.The events are independent 4.The number of times that the outcome of interest occurs in k events is designated as x. Probabilities of x designated as p(x)
Using the characteristics of the Normal Distribution answer the following question: The length of bluegill fish is normally distributed with a mean of 70mm and a standard deviation of 5mm. What percentage of bluegill fish is greater than 75mm? 34.13% 50% 68.26% 15.87%
15.87
Binomial Distribution
A probability model used with 2 possible outcomes (Discrete) Allergy medication works, or doesn't work Children w bacterial infection respond to antibiotic, or not Kittens born as male, or female Success Or Failure Number of trials defined
Which of the following are characteristics of the Normal Probability Distribution Mean Standard Deviation Asymptotic tails All of the above
All the Above
Why is it impossible to calculate a 100% confidence interval?
As confidence increases, Z continues to infinity. So there is no finite Z value for 100% confidence. The Normal Distribution is not bounded at the upper or lower end - it 'contains' infinite standard deviations. A calculation of a 100% confidence interval implies finite boundaries, and is therefore by definition of the normal distribution, an impossibility
Coefficient of Variation
C.V. is the s.d. divided by the mean and put into % terms (s.d. / Xbar) * 100 since it is in %, one can compare the variability between 2 or more groups, or years the bigger the %, the more variation
How was Normal Distribution discovered?
Discovered by Abraham de Moivre in 1733 when he was looking for a way to approximate binomial probabilities when the number of trials was large and p = 0.5. After graphing the data, he discovered a trend. his theorem was later proved for arbitrary values of p by Pierre Simon Laplace.
Point Estimate
Estimating the boundaries we expect with _____ confidence the "true" population mean to fall within
Nominal Scale or Classification Set
Events fall into a particular class or category of which there may be two or more categories
The correct equation to apply to the following problem is the Z formula for a sampling distribution: What is the probability that any randomly sampled individual mosquito fish (with a population mean of 34.29 and standard deviation of 5.49) would have a length of 40mm or larger? True False
False
Two events are independent when the occurance of the first event has an effect on the probability of the second event occuring. True False
False
The antelope band data is an example of variable measured on a continuous scale. True False
False-Correct there is not an infinite values between connected points (you can't have 1.5 antelope bands)
The Y axis of a histogram is the units of variable True False
False-The Y axis is the vertical axis and that is the frequency
Box and Whiskers Plot
Find the median Find the median of the "second half" Find the median of the "first half" Plot Q1, Q2, Q3 Draw a box around Q1- Q3 Draw whiskers to max and min 3.9, 4.1, 4.2, 4.3, 4.4, 4.4, 4.4, 4.4, 4.5, 4.5, 4.6, 4.7, 4.8, 4.9, 5.0, 5.1
Choose which term (below) best completes this sentence: As the level of confidence increases (say from 80% to 99%), the interval range __________. Decreases or gets smaller The range stays the same because the SE does not change if the population does not change Increases or gets larger It depends on the sample mean
Increase or gets larger: The Confidence interval is the mean + or - (Z*(SE)) Where Z* is the probability of the Z that corresponds with your confidence level. The higher the confidence the higher the Z* and therefore the interval size is larger.
Binomial
K=The event occurred a specified number of time Ex: A coin flipped 5 times, K=5 P= Probability of the outcome of interest occurring or success Q= Probability of failure: P-1 5!= 5*4*3*2*1
Parameters from the Law (Binomial)
Mean= K*P Variance = k * p * q Standard Deviation = (k * p * q)^.5
Which of the following is NOT a measure of dispersion: Median Variance Range Standard Deviation
Median
If you are examining the variable, "Lobster Sex (male or female)" which measure of central tendency would you use to best describe the center of the distribution? Mean Median Mode Variance
Mode Lobster gender is a categorical variable, you would use mode to describe the most frequently occuring category - male or female
Multiplication Rule "and" Rule
Pr (A and B) = Pr(A)*Pr(B) Ex: Probability of choosing a queen of hearts (4 / 52) * (13 / 52) = 52 / 2704 = 1 / 52
Addition Rule "OR" rule (Mutually Exclusive)
Pr (A or B) = Pr(A) + Pr(B) Ex: Probability of picking student A or B (1 / 100) + (1 / 100) = 2 / 100
Addition Rule (Dependence)
Pr (A or B) = Pr(A) + Pr(B) - Pr(A)*Pr(B) Pr(drawing a Queen or a Heart)= Pr(Queen) + Pr(Heart) - Pr(Queen)*Pr(Heart)= (4 / 52) + (13 / 52) - (4 / 52) * (13 / 52)= 17 / 52 - 1 /52 = 16 / 52 = 30.77%
As the sample size increases, the standard error of the mean gets: Larger Smaller
Smaller: As the sample size gets larger, the SE of the mean gets smaller, and therefore our estimates become more precise.
Which measure of dispersion is the most useful measure to describe the average dispersion of the sample? Standard Deviation Range Variance Either A or B
Standard Deviation-Yes, because the standard deviation is in the units of the measured variable it is the most useful measure of dispersion
If the measures of central tendancy are not all equal, this tells us....
The data is skewed
Z Score
The number of standard deviations FROM THE MEAN an observation lies
Conditional Probability
The probability that Event A occurs, given that Event B has occurred
Ordinal (Ranking) Scale
This is where less than and greater than come into play, or not improved, somewhat improved or greatly improved
A property of the normal probability distribution is that the distribution is completely defined by the mean (mu ) and standard deviation (sigma ). True False
True
T/F When k is fairly large and p is not too near 0 or 1, the binomial distribution becomes approximately a normal distribution. True False
True
The correct equation to apply to the following problem is the Z formula for a sampling distribution: What is the probability that a random sample of 10 mosquito fish (with a population mean of 34.29 and standard deviation of 5.49) would have a mean length of 40mm or larger? True False
True
A Statistic is from a Sample and a Parameter is from a Population True False
True - when you draw a sample from a population you will calculate statistics; but if you have data from the entire population your Mean, etc. is a parameter
The Central Limit Theorem states that even if a continuously measured variable is not normally distributed, the distribution of repeated sample means will be normally distributed. True False
True: In the video I show the example of city populations which is not normally distributed but if taken multiple samples, get the mean from each sample, plot the distribution of sample means THAT distribution is normally distributed.
One of the conditions of using or applying the binomial distribution is that the events must occur a specific amount of times. True False
True: k = a value. Take the kittens example, tossing a coin, each of these examples were given a specific # of events. The cat had 6 kittens, the coin was tossed 2 times.
Which measure of dispersion is in the squared units of the variable? deviation range variance standard deviation
Variance
Which of the following is NOT a measure of central tendency Mean Variance Median Mode
Variance
Type 2 Error
When you fail to reject the null hypothesis when you should have done so
Normal Distribution
based upon mean and standard deviation like the Binomial, there exists a never ending number of distributions
Interquartile Range
measures the range of the middle half of the data Q3 - Q1 eliminates extreme values
Fill in the blanks in the formula for the mean: xbar= Sigma*X/ ______
n
Relative Measures of Dispersion
s.d.'s are in measurement terms; i.e. quarts, mm, people, lobsters, degrees, etc. cannot compare people to lobsters using s.d.'s because of the units of measure; most people are bigger than rats
When constructing a confidence interval there are 2 components that will have a great impact on the size of the interval (upper limit - lower limit). What are these 2 factors? sample mean and confidence level sample standard deviation and confidence level sample mean and sample size sample size and confidence level
sample size and confidence level: as the sample size increases the standard error will become smaller and as the confidence level increases the value of Z will increase
Sum of Squares
squaring eliminates the negatives and the sum is a nonzero positive value often becomes a really big number
Which measure of dispersion is in the units of the variable? mean median sum of squares standard deviation
standard deviation is the square root of variance
Characteristics of Normal Distribution
symmetrical area under the curve sums to 1.0 X can take values; -∞ to ∞ asymptotic tails
Variance
the n-1 allows for sampling error and makes the statistic unbiased the "average" squared distance of all observations from the mean
Standard Deviation
the square root of the variance same scale as the mean! it is a measure of the "average" amount by which each observation in a data set differs from the mean
Conditionally Dependent
two events, X and Y, are conditionally dependent if the outcome of Y depends on X, or X depends on Y
Independent
two events, X and Y, are independent if the outcome of Y does not influence the outcome of X Ex: winning numbers on the roulette wheel
Mutually Exclusive
two events, X and Y, are mutually exclusive if the occurrence of one prevents the occurrence of the other. Also, the events can not occur at the same time. Ex: Sex, Male or Female or Rolling dice (you cant role a 2 AND a 6 at the same time)
Type 1 Error
when you reject the null hypothesis when you should not have done so.