Probability Density Curve
the curve used to describe the distribution of a continuous random variable, tell us what proportion of the population falls within any given interval
the area under the entire curve is equal to 1
in general the area under a probability density curve between any two values a and b has the interpretations:
it represents the proportion of the population whose values are between a and b, and it also represents the probability that a randomly selected value from the population will be between a and b
Normal Curve
type of probability density curve, have one mode, are symmetric around the mode, the mean and median of a normal distribution are both equal to the mode, in other words, the mean, median, and mode of a normal distribution are all the same
for any interval
Population Standard Deviation on a Normal Curve
standard deviation measures spread, so the normal curve is wide and flat when the population standard deviation is large, and tall and narrow when the population standard deviation is small
The Normal Distribution Follows the Empirical Rule
approx 68% of the population is within one standard deviation of the mean
95% within two standard deviations
99.7% within three standard deviations
Standard Normal Distribution
normal distribution that has mean 0 and standard deviation 1
Standard Normal Curve
the probability density function for the standard normal distribution
z-Score
since the mean of the standard normal distribution, which is located at the mode, is 0, the z-Score at the mode of the curve is 0. Points on the horizontal axis to the left of the mode have negative z-Scores, and points to the right of the mode have posit
The Notation Z(alpha)
Let alpha be any number between 0 and 1, the notation Z(alpha) refers to the z-score with an area of alpha to its right
Properties of the Z-Score
1) the z-score follows a standard normal distribution
2) values below the mean have negative z-scores, and values above then mean have positive z-scores
3) the z-score tells how many standard deviations the original value is above or below the mean
Finding Areas Under a Normal Curve
an area under a normal curve over an interval can be interpreted in two ways: it represents the proportion of the population that is contained within the interval, and it also represents the probability that a randomly selected individual will have a valu
Sampling Distribution of Sample Mean
the probability distribution of the sample mean
if several samples are drawn from a population, they are likely to have different values for sample mean, because the value of sample mean varies each time a sample is drawn, sample mean is a random variable
Population Proportion
in a population, the proportion who have a certain characteristic
denoted by p
Sample Proportion
in a simple random sample of n individuals, let x be the number in the sample who have the characteristics
p hat = x/n
denoted by p hat
Sampling Distribution
the probability distribution of the sample proportion
if several samples are drawn from a population, they are likely to have different values of sample proportion, because the value sample proportion varies each time a sample is drawn, sample proportion
The Central Limit Theorem for Proportions
havea a sample proportion for a sample size of n and population proportions p, if np less than or equal to 10 and n(1-p) less than or equal to 10 then the distribution of sample proportions is approximately normal
The region above a single point has zero width, and thus an area of 0
If X is a continuous Random Variable then P(X=a)= a for any number a
P(a < X < b) = P(a <= X <= b)
Standardization
method to convert x, a value from a normal distribution, to a z score
Have sample mean of a simple random sample of size n, drawn from a population with a mean and standard deviation
the mean of the sampling distribution equals the mean of the population
standard deviation of sampling distribution on sheet
If a sample size is large enough the sample mean with be:
approximately normally distributed
for a symmetric population the sample mean is approximately normally distributed even for a small sample size, for a skewed population the sample mean must be larger for the sample mean to be approximately normal
Central Limit Theorem
fact that the sampling distribution of the sample mean is approximately normal for a large sample from any distribution
the sample mean has an approximately normal distribution, with mean of the sampling distribution equal to the mean of the population, a
Have a Sample Proportion of a simple ransom sample size n, drawn from a population with population proportion p
the mean if the sampling distribution equals the population proportion
the standard deviation of the population proportion is on the sheet