The central limit theorem

what is the shape of the normal distribution for the population

distribution for population

z-score

the number of standard deviations that a particular X value is away from the mean.

If a large number of samples are selected from a normally distributed population or if a large number of samples that is greater than or equal to 30 are selected from a population that is not normally distributed, and the sample means are computed,

then the distribution of sample means will look like little distorted bell shape

what determines the spread or dispersion of a variable

the variance and standard deviation of a variable. larger the variance, the more the data values are dispersed.

Formula for the mean and standard deviation for the binomial distribution

μ= n*p σ= √(n*p*q)

the mean of the sample mean

μx̄= sum of x̄ divided by total frequency, n

standard deviation central limit theorem

σ/√n

σ/√n

σx̄ standard deviation of sample mean = standard error of the mean

When finding: 1. P(X=a) 2. P(X≥a) 3. P(X>a) 4. P(X≤a) 5. P(X<a)

Use: 1. P(a-0.5<x<a+0.5) 2. P(X>a-0.5) 3. P(X>a+0.5) 4. P(X<a+0.5) 5. P(X<a-0.5)

formula for the z value when adjusting large sample from small population

(x̄-μ)/(σ/√n)* √((N-n)/(N-1))

σx̄

(σ/√n)* √((N-n)/(N-1))

binomial distribution is determined by what factors. What is the relationship of factors

number of trials, n and the probability of success, p when n is large and p is closer to 0.5, the shape of the binomial distribution will be similar to the shape of the normal distribution.

adding or subtracting 0.5 from the data value is for

the correction factor

When the population is large and the sample is small,

the correction factor is not used, since it will be close to 1.00

sampling error

the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population

as the sample sizes get larger

the distribution of means calculated from repeated sampling will approach normality.

what happen to the shape of normal distribution when the value of standard deviation increases

the shape of the curve spreads out

graph of the sample mean

y axis as frequency, x axis as sample mean

which formula is used to gain information about a sample mean

z= x̄ -μ / (σ/√n)

z with large sample size, z score to gain info about a sample mean

z= x̄ -μ / (σ/√n)

x̄

sample mean

(σ/√n)

standard deviation of the sample mean

standard error of the mean

standard deviation of the sample means

why a normal distribution can be used as an approximation to a binomial distribution

when n is large and p is closed to 0.5, the distribution has a normal shape

when using the central limit theorem, if the original variable is not normal,

a sample size of 30 or more is needed to use a normal distribution to the approximate the distribution of the sample means. The larger the sample, the better the approximation will be.

standard normal distribution

normal distribution with a mean of 0 and a standard deviation of 1.

Third property of the sampling distribution of sample means

pertains to the shape of the distribution and is explained by the central limit theorem.

how to write a probability of any z value between 0 and 2.32

P(0<z<2.32)

____________correction factor

finite population

The formula, z= x̄ -μ / (σ/√n) is used to

gain information about a sample mean used to gain information when applying the central limit theorem about a sample mean when the variable is normally distributed or when the sample size is 30 or more.

The formula, z = x - μ / σ is used to

gain information about an individual data value obtained from the population when the variable is normally distributed.

According to Chebyshev's Theorem, at least what proportion of the data will be within 2 standard deviations?

at least three fourths, or 72 percent of the data will fall. the result is found by substituting k=2 in the expression, 1-1/k^2

what is the common shape of continuous variable

bell shaped distribution called approximately normally distributed variables.

how to find specific data value by given percentage.

by substituting the values for z, μ, and σ. x= z*σ +μ

what is the most important fact about a normal distribution?

central limit theorem

distribution of means for all sample of certain size from the population

distorted curve

normal distribution of individual data for the population

lower curve

Why standard deviation of sample mean is much less variable than the distribution of the individual data values.

as sample size increases, the standard deviation of means decreases.

characteristics for binomial distribution

1. There must be a fixed number of trials. 2. The outcome of each trial must be independent. 3. Each experiment can have two outcomes or outcomes that can be reduced to two outcomes. 4. The probability of a success must remain the same for each trial.

what are characteristics of a normal distribution curve?

1. bell shaped 2. continuous 3. symmetric

Central Limit Theorem. what is its mean and a standard deviation

As the sample size n 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 (Pop mean) and a standard deviation (pop standard deviation divided by square root of sample size)

How to transform the original variable to a standard normal distribution variable?

It is by using the formula, z=(value - mean)/standard deviation.

Chebyshev's Theorem

The proportion of 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. (k is not necessarily an integer)

Continuous variable and example

a continuous variable can assume all values between any two given values of the variables. Example: Height of the adult men, body temperature of rats. cholesterol level of adults

correction factor

√((N-n)/(N-1)) N= population size, n= sample size

σx̄

√((Σ (x̄ − μ)^2)/n (total frequency)

How to find data values for specific probalities

1. Draw a normal curve and shade the desired area that represents the probability. 2. Find the z value from the table that corresponds to the desired area. 3. Calculate the x value by using the formula, x= z*σ +μ

How can we find the area under the standard normal curve?

1. Draw a normal curve and shade the desired area. 2. Convert the values of X to z values, using the formula, z= (x̄ -μ / σ) 3. Find the corresponding area using a table.

Find the correction of each. 1. P(X is less than or equal to 7) 2. P(X=8) 3. P(X is greater than or equal to 3)

1. P(X is less than 7.5) 2. P (X is greater than 7.5 and less than 8.5) 3. P ( X is greater than 2.5)

what are the properties of the theoretical normal distribution.

1. a normal distribution is bell shaped. 2. The mean, median , and mode are equal and are located at the center of the distribution. 3. A normal distribution curve is unimodal 4. The curve is symmetric about the mean 5. The curve is continuous without gaps or holes. 6. The curve never touches x axis 7. The total area under a normal distribution curve is equal to 1.00 or 100% 8. The area under the part of a normal curve that lies within 1 standard deviation of the mean is 0.68 or 68%. Within 2 standard deviations, about 0.95, or 95%. within 3 standard deviations, about 0.997 or 99.7%

two properties of the distribution of sample means

1. the mean of the sample means will be the same as the population mean. 2. The standard deviation of the sample means 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.

According to Chebyshev's Theorem, at least what proportion of the data will be within 3 standard deviations?

At least eight ninths, or 88.89 % of the data value will fall within 3 standard deviations of the mean.

For all normal approximation to the binomial distribution, what is true?

For all cases, μ= n*p σ= √(n*p*q) n*p ≥ 5 n*q ≥ 5

When a distribution is positively skewed, the relationship of the mean, median, and mode from left to right will be

Mode, median, mean

what is the shape of a continuous distribution and a discrete distribution?

Normal distribution curve and binomial distribution.

definition of z value or z score

the number of standard deviations that a particular X value is away from the mean

the mean of the sample mean will be the same as the population mean. if

all the possible samples are selected with replacement of the population.

correction for continuity

a correction employed when a continuous distribution is used to approximate a discrete distribution.

State the empirical rule

about 68% of the area lies within 1 standard deviation of the mean, about 95 percent within 2 standard deviation and about 99.7 percent within 3 standard deviations of the mean.

What shape of distribution is needed for application of Chebyshev's theorem

any distribution

Sampling distribution of sample mean

distribution using the means computed from all possible random samples of a specific size taken from a population

what is empirical rule?

empirical rule states that when a distribution is bell shaped, then * 68 percent of the data values will fall within 1 standard deviation of mean. * 95 percent of the data values will fall within 2 standard deviations of the mean. * 99.7 percent of the data values will fall within 3 standard deviations of the mean.

The correction factor is necessary if,

if relatively large samples are taken from a small population, because the sample mean will then more accurately estimate the population mean and there will be less error in the estimation.

draw negatively skewed and positively skewed distributions

negatively skewed: mean, median, mode<Right> positively skewed : mode, median, mean <Left>

condition when np is greater than or equal to 5 and nq is greater than or equal to 5 is when

normal distribution can be used as an approximation

Shape of binomial distribution depends on

number of trials, n and the probability of success, p q=1-p when n is large and p is closer to 0.5, the shape of the binomial distribution will be similar to the shape of the normal distribution. but when p is close to 0 or 1 and n is small, normal approximation is inaccurate. a normal approximation should be used only when n*p and n*q are greater than or equal to 5.

draw normal distribution which has same mean but different standard deviation . and the one which has different means but same standard deviation.

p313 answer in the book

σ

sigma; symbol for standard deviation of the population, √((Σ (X − μ)^2)/N) N= size of population

procedure for the normal approximation to the binomial distribution

step 1. Check to see whether the normal approximation can be used step 2. Find the mean μ and the standard deviation σ step 3. write the problem in probability notation, using X step 4. Rewrite the problem by using the continuity correction factor, and show the corresponding area under the normal distribution. Step 5. Find the corresponding z values. Step 6. Find the solution.

when drawing normal distribution , why we often skip the y axis?

the are under a normal distribution curve is used more often than the values on the y axis.

In central limit theorem, larger the sample

the better the approximation of the distribution of the sample means.

When using the central limit theorem, if the original variable is normally distributed,

the distribution of the sample means will be normally distributed for any sample size n

The shape and position of a normal distribution depend on two parameters which are...

the mean and standard deviation

If all possible samples of size n are taken with replacement from the same population,

the mean of the sample means equals the population mean. The standard deviation of the sample means equal to population standard deviation divided by sample size.

Correction for continuity is necessary because

the normal distribution is continuous and the binomial distribution is discrete.

the standard deviation of all possible sample means equals

the population standard deviation divided by the square root of the sample size

The formula for the standard error of the mean (σ/√n) is accurate when

the samples are drawn with replacement or are drawn without replacement from a very large or infinite population.

when do we need a correction factor?

the sampling with replacement is unrealistic. it is necessary for computing the standard error of the mean for samples drawn without replacement from a finite population.

To adjust large samples taken from a small population,

the standard error of the mean must be multiplied by the correction factor (σ/√n)* √((N-n)/(N-1))

when can we use the empirical rule

when a distribution is bell shaped, the empirical rules are true. 68 (1s) -95 (2s) -99.7(3s)

Z score for individual

z = x - μ / σ

mean central limit theorem

μ