Statistics 2
What is the probability of an exact z value within a continuous distribution?
0 since a vertical line has no area
In regards to the Empirical Rule, when the spread of data is within 68% from the mean, what is the standard deviation?
1
What are the characteristics of a normal distribution?
1. Curve is bell shaped 2. Mean, median, and mode are equal and located at the center of the distribution 3. Curve is unimodal 4. Curve is symmetric 5. Curve is continuous 6. Curve never touches the x-axis 7. Total area under distribution is 1.0 OR 100% 8. Area 1 standard deviation from the mean is 0.68 OR 68%; Area 2 standard deviation from the mean is 0.95 OR 95%; Area 3 standard deviation from the mean is 0.997 OR 99.7%
What are the steps in determining normality?
1. Draw histogram and determine shape 2. Determine the Pearson coefficient; if index is greater than or equal to +1 OR less than or equal to -1, then data are significantly skewed 3. Determine if there are any outliers
The shape and position of a normal distribution curve is dependent on what two parameters?
Mean Standard deviation
When majority of the data values fall to the right of the mean, what is the distribution?
Negatively or left skewed distribution
What is the formula in order to determine the pearson coefficient?
PC = 3(mean-median)/std deviation
Define sampling error
The different between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population
Define Chebyshev's theorem
The proportion of data values from a data set that will fall within k standard deviations of the mean will be at least 1- (1/k^2), where k is a number greater than 1
In regards to Chebyshev's theorem, when the spread of data is within 89% from the mean, what is the standard deviation?
3
In regards to the Empirical Rule, when the spread of data is within 99.7% from the mean, what is the standard deviation?
3
What is another name for the Empirical Rule, which also defines the Empirical Rule based on the name?
68-95-99.7 Rule
Describe the phrase 'sampling distribution of sample means'
A distribution using the means computed from all possible random samples of a specific size taken from a population
What are the properties of the distribution of sample means?
1. The mean of the sample mean will be the same as the population mean 2. The standard deviation of the sample mean will be smaller than the standard deviation of the population, and it will be equal to the population standard deviation divided by the square root of the sample size 3. The central limit theorem
What are the two stipulations to remember when utilizing the central theorem?
1. When the original variable is normally distributed, the distribution of the sample means will be normally distributed 2. When the distribution of the original variable is not normal, a sample size of 30 or more is needed to use a normal distribution to approximate the distribution of the sample means. The larger the sample, the better the approximation will be.
In regards to Chebyshev's theorem, when the spread of data is within 75% from the mean, what is the standard deviation?
2
In regards to the Empirical Rule, when the spread of data is within 95% from the mean, what is the standard deviation?
2
What is the central limit theorem?
As the sample size increases without limit, the shape of the distribution of the sample means taken with replacement from a population with mean and standard deviation will approach a normal distribution. This distribution will have a mean and a standard deviation that is computed by population standard deviation divided by the square root of sample size
What is the distribution shape of continuous variables?
Bell shaped
How do you find the area under a normal distribution, which is to the left of any z value?
Look up the z value on the table, and use the area given
Define standard normal distribution
Is a normal distribution with a mean of 0 and a standard deviation of 1
What is the significance of a z score?
It demonstrates how many standard deviations (distance) a value is from the mean
What is a normal distribution?
It is when a random variable has a probability distribution whose graph is continuous, bell shaped, and symmetric
How do you find the area under a normal distribution, which between any two z values?
Look up both z values and subtract the corresponding areas
How do you find the area under a normal distribution, which is to the right of any z value?
Look up the z value and subtract the area from 1
When majority of the data values fall to the left of the mean, what is the distribution?
Positively or right skewed distribution
What are fractiles?
Same as quantiles
In regards to the correction factor for finite populations, what is the formula?
Sample mean subtracted by the population mean divided by standard deviation divided but he root of sample size multiplied by the root of population size subtracted by the sample size divided by the population size subtracted by 1
What are quantiles?
Separates data sets into approximately equal groups
What is the difference between Chebyshev's theorem and the Empirical Rule?
The Empirical Rule is used when data distribution is bell shaped, whereas Chebyshev's theorem is used for all distribution shapes
How do you figure out the probability distribution curve?
The same way you figure out the area
What is a standard error of mean?
The standard deviation of sample means
How do you compute the z score?
Value - Mean / Standard Deviation
How do you find data values when given probabilities?
X = (z) times std deviation + mean
What determines normality in regards to a normal quantile plot?
points lying in a straight line
How do you compute the standard deviation of sample means?
standard deviation divided by the root of the sample size
What is the rounding rule in regards to z values?
two decimal places
Within a normal quantile plot, what values are recorded on the x-axis? What about the y-axis?
x-axis data values y-axis z values