Statistics Project

Statistics Project 1

Fashion designers:

Introduction:

This paper analysis the income levels of employees in the fashion designers industry, this industry according to the bureau of labour in the United States it is estimated that this industry employs over 20,000 individuals according to the year 2006 statistics. This industry mainly focuses on dress making, clothing, shoes of different styles and making.

Data on the income levels of employees in the fashion industry was retrieved from the bureau of statistics in the US which is available at www.bls.gov .

The data:

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Data was retrieved from http://www.bls.gov/oes/current/oes271022.htm where states which include new York, Oregon, new jersey, California, Massachusetts and Missouri, the data contains employment levels in these states, hourly wage rate and the mean annual income in terms of wage, the data below shows the data:

State

Employment

Hourly mean wage

Annual mean wage

New York

7,170

$37.88

$78,800.00

California

3,550

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$32.83

$68,290.00

Massachusetts

660

$28.59

$59,470.00

Oregon

220

$30.83

$64,120.00

New Jersey

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470

$32.62

$67,860.00

Missouri

110

$32.51

$67,630

The above figures can be represented in a bar graph as follows:

The chart below shows the level of employment in the selected states:

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The chart bbelow shos the level of hourly wage in the states selected:

The chart below shows the level of annual wage for the states selected:

Mean, standard deviation and median:

When we use ungrouped data to analyse the mean and the median of the data our results are as follows:

State

Employment

Hourly mean wage

New York

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7,170

$37.88

California

3,550

$32.83

Massachusetts

660

$28.59

Oregon

220

$30.83

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New Jersey

470

$32.62

Missouri

110

$32.51

total

12,180

$195.26

mean

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2,030

$32.54

min

$28.59

max

$37.88

range

$9.29

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standard deviation

3.070294231

The mean hourly wage is 32.54 dollars, the range is 9.29 and our standard deviation is equal to 3.07, these are measures of central tendencies of data, the mean gives us an estimate of the hourly wage rate in the fashion industry and the standard deviation give us the measure of deviations from the mean of the different wages paid by different states.

Grouped data:

When we group the data into 5 classes and considering the class interval to be two then we will be in a position to obtain our frequency and therefore construct a histogram, after grouping our data the results are as follows:

class

frequency

cumulative  frequency

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percentage

cumulative  percentage

28.50 to30.50

1

1

17%

17%

30.51 to  32.50

1

2

17%

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33%

32.51 to  34.50

3

5

50%

83%

34.51 to  36.50

0

5

0%

83%

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36.51 to  38.50

1

6

17%

100%

total

6

100%

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Our histogram will be as follows:

This histogram shows that there is a high possibility that the wage rate will be between 32.51 to 34.50, to be precise the probability that the wage rage will be at this level is 0.5 or 50% probability.

Also our or give will be as follows:

The orgive represents the cumulative frequency data and shows the trend of the cumulative frequency to the 100% level.

The stem and leaf:

A stem and leaf diagram displays the trends in data and also gives us an overview of the nature of the data, whether skewed or normal distribution. Below is the stem and leaf diagram:

Stem and leaf

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28 59

30 83

32           51            62            83

37           88

The above is the stem and leaf representation of the data, it is clear that moist of the observation are in the wage rate 32, this data therefore is skewed to the left and does not assume a normal distribution.

Binomial probability distribution:

The binomial probability distribution is applied to find the probability that an outcome will occur in a given number of trials. The variable in this case however must be a discrete dichotomous random variable, in this distribution we consider n identical trials, each trial has two possible outcomes where we refer to a success and the other as a failure, a success in our case will be denoted as P and a failure will be denoted as Q. finally the outcome of one trial does not affect the outcome of the other trial,

In our case we will construct the binomial probability distribution using the statement that the employment level in the fashion and design industry is expected to grow by 5%, assuming that our level of employment in our selected states is 12,000 then we expect that in 2016 the

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employment level will be 70,000.

According to this statistics the employment level is based on a 2006 report and therefore the time period is 10 years, which also means 120 months, so employment level is expected to increase by 5 individual each month. This statistics were retrieved from http://www.bls.gov/oco/

oco2001.htm#emply . if now we assume that the probability of this happening is 70% then our binomial probability distribution will be as follows:

The binomial probability function is given by:

P (x) =    n           ∏ x ( 1-∏ ) n-x

X

Where in our case n = 5 which is the number of employment per month, x = 0,1,2,3,4,5) which are the number of outcomes per month, ∏ = 0.7 which is the probability that the employment level will increase by 5% from 2006 to 2016.

Our binomial distribution is as follows after calculations:

x

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P(x)

0

0.00243

1

0.02835

2

0.1323

3

0.3087

4

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0.36015

5

0.16807

If we are to draw a chart regarding the binomial probability distribution then our chart will be as follows:

The binomial probability distribution helps us estimate the probability of an outcome, in this case we can be in a position to estimate the probability for example what is the probability that the persons who are likely to be employed will be greater than 2 individuals, more than 3 individuals or even less than one individuals, for this reason therefore the probabilities can be calculated by adding the probabilities of each outcome to come up with the desired answer in question.

Hypothesis testing:

We still consider our data from the fashion design industry to analyse the data, in hypothesis testing we will consider hypothesis test for the data and stating the null and alternative hypothesis, in this case therefore it is clear that we will have to use the T table, Z table or even the F table on the nature of the test and deepening on the hypothesis in question

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Confidence interval:

90% confidence interval:

When we are constructing the confidence interval we consider the standard deviation, the mean and the value from the T tables at 90% level of measure: we lookup 10% at two tail from the T table and the figure is 2.015048:

Our confidence interval will take the following form:

P(x – st) ≤ (x + st) = 90%

Where X is the mean, S is the standard deviation and T is the value from the tables:

P(32.54 –(3.07 X 2.015) ≤ X ≤ (32.54 + (3.07 X 2.015) = 90%

P(26.35395) ≤ X ≤ (38.72605) = 90%

This confidence interval states that at 90% confidence interval the mean will range from 26.35 to 38.72 where they are the lower and upper bound respectively. This also means that we are 90% confident that the mean ranges from 26.35 to 38.72

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95% confidence interval:

When we lookup 5% at two tail t test then the value is 0.726687, therefore our confidence interval will be as follows:

P(32.54 –(3.07 X 0.726687) ≤ (32.54 + (3.07 X 0.726687) = 95%

P(30.30907091) ≤ X ≤ (34.77093) = 95%

This confidence interval states that at 95% confidence interval the mean will range from 30.30 to 34.77 where they are the lower and upper bound respectively. This also means that we are 95% confident that the mean ranges from 30.30 to 34.77.

From the measure of confidence interval it is clear that when we consider a larger confidence interval then it is clear that the lower is the range of the interval as compared to when we use a lower confidence interval.

Linear regression:

We will perform the regression model on the employment level and the hourly wage rate, we will

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assume that the higher the level of employment then the higher is the wage rate, therefore we will assume that the wage rate dependent on the rate of employment, in this case therefore our dependent variable will be wage rate and the independent variable will be employment level:

State

Employment

Hourly mean wage

y2

x2

xy

New York

7170

37.88

1434.894

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51408900

271599.6

California

3550

32.83

1077.809

12602500

116546.5

Massachusetts

660

28.59

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817.3881

435600

18869.4

Oregon

220

30.83

950.4889

48400

6782.6

New Jersey

470

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32.62

1064.064

220900

15331.4

Missouri

110

32.51

1056.9

12100

3576.1

total

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12180

195.26

6401.545

64728400

432705.6

mean

2030

32.54333333

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After estimation our:

B = 0.0005673

α = 31.391809

Therefore our estimated model will take the following form below:

Y = 31.39 + 0.0005673 X

We can define this model as follows, if we hold all other factors constant and the level of employment is zero then the level of wage rate will be 31.39. if we hold all other factors constant and increase the level of employment by one unit then the wage rate level per hour will increase by 0.0005673 units.

For this reason therefore it is clear that our earlier stated objective has been achieved, this is in reference to the objective that an increase in employment will raise the wage rate level.

Correlation:

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When we undertake the calculation of the Pearson correlation coefficient then our correlation after calculation is equal to 0.8366, from the figure of the coefficient it is clear that we have positive correlation between the two data, we also have a moderately strong relation and this is obtained by the fact that the correlation coefficient is close to 1, we therefore can conclude that there is a strong positive correlation between employment and wage rate per hour.

Summary:

From our statistical analysis that we have performed on the fashion and design industry it is clear that the industry provides employment to a large number of individuals in the United

States , in our selected states which are 6 in number the industry employs over 12,000 individuals according to the 2006 statistics.

According to the bureau of labour in the United States the growth rate of this industry is expected to grow by 2016 where its employment rate will increase by 5%, when calculating using the percentage given then it is clear that by 2016 the employment level of the industry in our selected state will increase from 19,000.

When we perform a linear regression estimation of the data and consider that the wage rate is dependent on the employment level then it is clear that the employment level positively affect the wage rate, this is to say that the higher the employment level then the higher is the wage rate. Further we found a strong correlation coefficient between wage rate and employment.

Finally we conclude by saying that there is a need to use a larger sample size in order to get a clearer picture of the fashion and design industry, a large data sample will allow us to overcome biasness in statistical analysis, samples are expected to be a representative of the entire population, for this reason therefore there is need to select a larger sample size and compare

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the results.

References:

Burbidge (1993) Statistics: An Introduction to Quantitative Research,

McGraw Hill, New York

Kroenke (1997) Data Processing: Fundamental, Design and Implementation, Prentice Hall publishers, New York

United States bureau of statistics (2008) the fashion design industry, retrieved on 9th January, available at

http://www.bls.gov/oes/current/oes271022.htm

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