Shelly K Bernard
MTH332-01
Assignment 3

PROBLEM A:
Using excel, generate a list of random 100 1s or 0s with the probability of 1s being prob=0.3

=If (rand( )<0.3,1,0)

This command means generate a uniformly random list with 1s and 0s where 1s appear 1/3 of the time; I have this list printed from J2 across to 102 on my excel sheet.

0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0

PROBLEM B:
From the data generated in a, calculate a 95% confidence interval for the proportion of 1s in the sample.

The confidence interval is [a, b] such that

and

Where p is printed in column G in my excel worksheet and it is the p-sample/sample-proportion = which calculates how many 1s out of the 100 values in the randomly generated list; z printed in column H is the number of standard deviations you have to be away from , the mean, to have 95% of the population under the distribution curve =(NORMINV(0.05/2, 0, 1); and s(p) printed in column I is the estimated-value/standard-error =.

prob= 0.3
95% CONFIDENCE INTERVAL
[0.174029268 0.345970732]

p-sample
0.26

z
1.959963985

s(p)
0.043863424

PROBLEM C:
Generate a large number of at least several thousand randomly generated 1ists of 1s and 0s. Then using a fixed confidence interval from a, calculate the proportion of lists for which the SAMPLE PROPORTION of 1s lies in that confidence interval.

LIST1:1000=If(AND (a<=p-sample, p-sample<=b), 1, 0) is the command that I used in column E3 to E1001 which means if the population sample is within the fixed confidence interval [a,b] from exercise a, then print 1, otherwise print 0. Then we will be able to calculate whether any of the thousands of lists’ sample proportions exist within the fixed confidence interval from exercise a 95% of the time.

To calculate the percentage of which SAMPLE PROPORTIONS exist within the fixed confidence interval, in cell E2 I used the command:

=

I found that for the 1,000 lists of 1s and 0s, each sample proportion lied within the fixed confidence interval from exercise a only 82.8% of the time!

PROBLEM D:
For each of your several thousand lists of 100 1s or 0s, calculate a separate 95% confidence interval. Calculate the proportion of these cases for which the TRUE PROPORTION 0.3 lies in the calculated confidence intervals.

prob= 0.3
95% CONFIDENCE INTERVAL
[0.174 0.346]
[0.257 0.443]
[0.165 0.335]
[0.219 0.401]
[0.285 0.475]
[0.174 0.346]
…

p-sample
0.26
0.35
0.26
0.35
0.26
0.35
…

z
1.959963985
1.959963985
1.959963985
1.959963985
1.959963985
1.959963985
…

s(p)
0.044
0.048
0.043
0.046
0.049
0.044
…

I created 10,000 1s and 0s in columns from J2 across to 102 and rows J2 down to J1002. To calculate the percentage of the number of times the TRUE PROPORTION 0.3 lies in the calculated confidence intervals I create another list in which the command states:

IF(AND(a<=0.3, 0.3<=b), 1, 0)

Which says if 0.3 lies within the confidence interval [a,b] print a 1, otherwise print 0. Then I calculated the percentage by adding up the number of 1s present and dividing by n=1000 and printed the result in cell F1.

PROBLEM E:
Repeat this for at least one other value of prob not equal to 0.3

prob= 0.9
95% CONFIDENCE INTERVAL
[0.174 0.346]
[0.156 0.324]
[0.257 0.443]
[0.148 0.312]
[0.201 0.379]
[0.192 0.368]
…

p-sample
0.26
0.24
0.35
0.23
0.29
0.28
…

z
1.959963985
1.959963985
1.959963985
1.959963985
1.959963985
1.959963985
…

s(p)
0.044
0.043
0.048
0.042
0.045
0.050
…

CONCLUSION:
In conclusion, I found that sometimes the true proportion will lie in the confidence interval, but that doesn’t happen a lot of the time, only 5% of the time. When I created the other thousand lists I found that each of the sample proportions will NOT lie in the 1st interval 95% of the time because the other sample proportions are independent from the first; they each have their own confidence interval. In essence, you actually don’t know what the next interval will contain after creating another list of 1s and 0s. You don’t know whether or not it will contain .3 so 95% ends up telling us nothing!

]]> http://missreid.wordpress.com/2008/04/18/assignment-3/feed/ 0 missreid http://missreid.wordpress.com/2008/04/15/assignment-2/ http://missreid.wordpress.com/2008/04/15/assignment-2/#comments Tue, 15 Apr 2008 16:23:41 +0000 missreid http://missreid.wordpress.com/?p=7 ]]>

Shelly K Bernard
MTH332-01
Assignment2

Definitions:

 x: a data entry
n: total number of data entries
μ and : mean of x
σ and s: standard deviation of x

Equations:

Mean:

The term “mean” or “arithmetic mean” is preferred in mathematics and statistics to distinguish it from other averages such as the median and the mode. In mathematics and statistics, the (arithmetic) mean of a list of numbers is the sum of all the members of the list divided by the number of items in the list. Sample mean is typically denoted with a horizontal bar over the variable x; enunciated as “x bar”.

Median:

In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one.

median

Skewness:

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. positive skew: The right tail is longer; the mass of the distribution is concentrated on the left of the figure. The distribution is said to be right-skewed. negative skew: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. The distribution is said to be left-skewed.

Kurtosis:

In probability theory and statistics, kurtosis (from the Greek word kyrtos or kurtos, meaning bulging) is a measure of the “peakedness” of the probability distribution of a random variable and how outlier-prone a distribution is. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations. The kurtosis of the normal distribution is 3. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 3; distributions that are less outlier-prone have kurtosis less than 3.

kurtosis

Uniform Distribution (continuous):

In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution’s support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. The distribution is often abbreviated X~U(a,b).

A continuous random variable X which has probability density function given by:

For as f(x) has a value of 1 on interval [0,1].

Normal Distribution:

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by two parameters, location and scale—the mean and variance, standard deviation squared —respectively. The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due to the central limit theorem.

The normal distribution also arises in many areas of statistics. For example, the sampling distribution of the sample mean is approximately normal, even if the distribution of the population from which the sample is taken is not normal. In addition, the normal distribution maximizes information entropy among all distributions with known mean and variance, which makes it the natural choice of underlying distribution for data summarized in terms of sample mean and variance. The normal distribution is the most widely used family of distributions in statistics and many statistical tests are based on the assumption of normality.

In probability theory, normal distributions arise as the limiting distributions of several continuous and discrete families of distributions. This distribution is often abbreviated X~N(, ). A normal distribution with mean value of 1 and standard deviation equal to 2 is given by:

 For .

 PROBLEM 1:
Using a moderate size data set, use R to complete the following tasks.

Data entered into R:

Rate of marriages per 1,000 total population residing in the United States in 2004.

x<-c(9.4, 8.5, 6.6, 13.4, 4.8, 7.4, 5.8, 6.1, 4.5, 9.0, 7.8, 22.8, 10.8, 6.1, 7.8, 6.9, 7.0, 8.8, 8.0, 8.5, 6.9, 6.5, 6.1, 6.0, 6.1, 7.1, 7.5, 7.1, 62.4, 8.0, 5.8, 7.4, 6.8, 7.3, 7.0, 6.6, 6.5, 8.1, 5.9, 7.6, 8.2, 8.4, 11.4, 7.9, 10.0, 9.4, 8.3, 6.5, 7.5, 6.2, 9.4)

sort(x)

Mean:

mean(x)= 8.939216

Median:

median
median(x)= 7.4

Skewness:

skewness(x) = 5.91365

Kurtosis:

kurtosis
kurtosis(x)= 39.08219

Histogram:

See Histogram

d

PROBLEM 2:
Use R to generate 1,000 uniformly distributed real numbers between 0 and 1, using the runif command.

x=runif(1000, 0, 1)

Mean:

mean(x) = 0.5069946

Median:

median
median(x) = 0.4988877

Skewness:

skewness(x) = -0.01528395

Kurtosis:

kurtosis
kurtosis(x) = 1.867278

Histogram:

See histogram

d

PROBLEM 3:

Using R to generate 1,000 normally distributed real numbers with mean 1 and standard deviation 2, using rnorm command.

x=rnorm(1000, 1, 2)

Mean:

mean(x) = 0.9273835

Median:

median
median(x) = 0.921323

Skewness:

skewness(x) = 0.0870111

Kurtosis:

kurtosis
kurtosis(x) = 2.720586

Histogram:

See histogram

Shelly K Bernard
MTH332-01
Assignment 1

 
Definitions:

x: a data entry
n: total number of data entries
µ and : mean of x
σ : standard deviation of x

Equations:
Mean:

The term “mean” or “arithmetic mean” is preferred in mathematics and statistics to distinguish it from other averages such as the median and the mode. In mathematics and statistics, the (arithmetic) mean of a list of numbers is the sum of all the members of the list divided by the number of items in the list. Sample mean is typically denoted with a horizontal bar over the variable x; enunciated as “x bar”.

Standard Deviation:

According to Wikipedia, the standard deviation of a probability distribution, random variable, or population/multiset of values is a measure of the spread of its values. It is usually denoted with the letter \sigma (lower-case sigma). It is defined as the square root of the variance, where the variance is the average of the squared differences between data points and the mean.

Z-score:

In statistics, a standard score is a quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing. The standard score indicates how many standard deviations an observation is above or below the mean. It allows comparison of observations from different normal distributions, which is done frequently in research. Standard scores are also called z-values, z-scores, normal scores, and standardized variables.

z-score

Problem Assignment:
Using any data set, use R to complete the following tasks

Data entered into R:

This data is a representation of the percentage of 15 to 24 year olds who dropped out of high school from 1990 to 2001 which is defined in the report as an “even drop out rate.”

4.0 4.2 4.3 4.4 4.5 4.8 4.9 4.9 5.0 5.1 5.5 5.8

Histogram:

Click to see Histogram

Mean:

mean(x) = 4.783333

Standard Deviation:

sd(x) = 0.5339958

Z-Score:

z-score
= -1.46692789 -1.09239311 -0.90512572 -0.71785833 -0.53059094 0.03121123 0.21847862 0.21847862 0.40574601 0.59301340 1.34208296 1.9038851