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The 195 F Coffee Company was established on February 23, 1990. The first 195 F Coffee store debuted in Singapore's Duxton Hill. James Morrisey founded the company in Ethiopia, which produces some of the world's greatest coffee beans. The company was named 195 F Coffee because the ideal temperature to make coffee was, and still is, 195 degrees Fahrenheit. This is a small firm, but it is well-known for its special brew coffee; it employs over 600 people in Singapore and Malaysia. They sell a variety of pastries in addition to coffee.

The 195 F Coffee Company has decided to expand its coffee shop area in Singapore and Malaysia.Establishment of more coffee shops calls for a huge start-up capital yet there is no guarantee that all of the coffee shop will make profit. In order to make the business more effective, me and my team has been appointed to be their advice manager. The purpose of this report is to provide a clear data analysis of 195˚F Coffee company. We will provide every data in detail and what should the company do to make it more efficient.

3. Measures of Central Tendency

Measures of central tendency, according to (), refer to statistical functions used to gauge the most typical, or highly expected, properties of a given study phenomenon. An investor studying a company’s stock prices, for example, will want to know the most typical, or conventional, characteristics of these prices so as to weigh if it is truly worth it investing into that particular company. It is only the measures of central tendency that provide a proper quantitative analysis that enable one to comprehend a phenomenon’s most conventional statistical characteristics. As () asserts, measures of central tendency, more or less, describe the typical statistical properties of a given variable hence they are also known as descriptive statistics. These measures encompass a wide array of statistical models namely: i) Mean ii) Median iii) Standard Deviation and iv) Interquartile Range. Below is an elucidation of these models and how the study uses them to describe typical (central) characteristics of the 195˚F Coffee company.

3.1 Pictorial Graph

In order to perform a decent data analysis, a statistics audit team was appointed to record the 195˚F Coffee company’s recent business activities. Several data variables were noted down such as the sample data for the food preparation time (see Appendix A) as well as the customers’ behaviour data (see Appendix B). A pictorial scatter graph is, thus, plotted below to help compute the four aforementioned measures of central tendency of these variables. Two variables are chosen for graphing: Time taken to prepare Hot Chocolate – which is the independent variable hence is plotted on the X-axis; and the Time customer spends in coffee shop – which is the dependent variable hence is plotted on Y-axis ().

Figure 1: Graph of Customer Behavior vs. Coffee Shop's Food Preparation

3.2 Mean

The mean, as () explains, is the most common measure of central tendency used to approximate the average of values of given variables. The mean is used in this study to approximate the average number of minutes the customers spend in the coffee shop based on the amount of time the staff take in food preparation. Below is the formula utilized in attaining this mean average:

Where: 1. = mean

X1, X2, X3 … Xn = intercept readings along the x-axis

n = number of total observations

Thus: = 25.05 + 21.28 + 18.43 + 24.57 + 20.92 + 19.68 + 20.37 + 10.59 + 16.44 +9.81 + 23.95 + 15.49 + 17.29 + 24.21 + 12.87 + 28.23 + 27.15 + 19.95+ 18.51 + 24.21 + 20.93 + 19.33 +25.48 + 22.79 + 29.26 + 16.43 + 10.36 + 15.86 +14.19 + 28.3

30

= 601.93 = 20.06433 = 20.06 minutes (rounded off to two decimal places)

30

Hence, the average time the customers spend at 195˚F Coffee Company is 20 minutes.

3.3 Median

() describes the median as the value that lies in the middle of a given dataset. As () asserts, the median is somewhat more robust than the mean because it is not influenced by outliers. Outliers are the values or integers that stand at the extreme end of a given dataset. Take for example a dataset that has the following values: -10, 12, 16, 23, 29, 22, 18, 14, 72. In this dataset, -10 and 72 are considered to be outliers because they are not within the same range as the other integers (). Such extreme outliers affect the true value of a mean measurement and may even give skewed averages. To avoid this, the median is incorporated to cushion the effect of outliers (median is not susceptible to outliers) (). For this current study, the median is used to find the true middle value of time spent by customer in the coffee shop as the mean calculated above may have been affected by outliers in the dataset. Below is the formula for computing the median:

Hence, median = 30 + 1 th item = 31 = 15.5th item

2 2

However, in order to find the 15.5th item, we have to find the average of the 15th and 16th item:

Reading from Figure 1 above, the 15th item from the graph is 12.87 while the 16th item is 28.23

Hence: 12.87 + 28.23 = 41.1 = 20.55

2 2

Hence, the median (middle) amount of time customers spend at 195˚F Coffee Company is 20.55 minutes.

3.4 Standard Deviation

According to (), standard deviation refers to statistical function that measures how far various data values are from the central mean. () says that it is crucial to calculate the standard deviation so as to know how to cater for those data values that fall far away from the mean. Take for example a social revolutionary initiative whose lead manager wants to purchase shoes for all students in a particular school. He has taken everyone’s shoe size measurement and calculated the mean to be shoe size 38. However, he has also discovered that several students have slightly different shoe sizes like 34, 36, and 40 (these values which are slightly different than the mean are the standard deviations). By knowing these different shoe sizes (standard deviation measurements) for several pupils, the lead manager will have avoided making the mistake of buying shoe size 38 for all students, and instead also buy shoe sizes 34, 36, and 40. For the current study, computation of the standard deviation measurements is done so as to know the other different amounts of time that customers spend at 195˚F Coffee Company besides the average 20.06 minutes. Below is the formula for standard deviation:

Where, x = intercepts along the x-axis as demonstrated in Figure 1 above

= average mean

n = number of observations

s = standard deviation

Σ = total summation of values

Hence:

s = (25.05 – 20.06)2 + (21.28 – 20.06)2 + (18.43 – 20.06)2 + (24.57 – 20.06)2 + (20.92 – 20.06)2 + (19.68 – 20.06)2 + (20.37 – 20.06)2 + (10.59 – 20.06)2 + (16.44 – 20.06)2 + (9.81 – 20.06)2 + (23.95 – 20.06)2 + (15.49 – 20.06)2 + (17.29 – 20.06)2 + (24.21 – 20.06)2 + (12.87 – 20.06)2 + (28.23 – 20.06)2 + (27.15 – 20.06)2 + (19.95 – 20.06)2 + (18.51 – 20.06)2 + (24.21 – 20.06)2 + (20.93 – 20.06)2 + (19.33 – 20.06)2 + (25.48 – 20.06)2 + (22.79 – 20.06)2 +( 29.26 – 20.06)2 + (16.43 – 20.06)2 + (10.36 – 20.06)2 + (15.86 – 20.06)2 + (14.19 – 20.06)2 + (28.3 – 20.06)2

s= 24.9 +1.49 +2.66 +20.34 +0.74 + 0.14 +0.1 +89.68 +13.1 +105.06 +15.13 +20.88 +7.67 +17.22 +51.7 +66.75 +50.27 +0.01 +2.4 +17.22 +0.76 +0.53 +29.38 +7.45 +84.64 +13.18 +94.09 +17.64 +34.46 +67.9 = 857.50

s= 857.50 = = = 5.4377 = 5.44 (rounded off to two decimal places).

30 – 1 29

To fully exploit the standard deviation measurement, we have to use the rule of 68-95-99.7 rule, which holds that:

1. 68% of observations fall within 1 standard deviation (σ) of the mean μ

2. 95% of observations fall within 2 standard deviations (σ) of the mean μ

3. 99.7% of observations fall within 3 standard deviations (σ) of the mean μ

Below is the formula for interpreting this rule:

μ + (n)σ

μ - (n)σ

where: μ = mean

n = number of standard deviations (either 1, 2, or 3)

σ = calculated standard deviation

Therefore:

1. For the 1st standard deviation:

= 20.06 + 1(5.44) = 25.5

= 20.06 – 1(5.44) = 14.62

Thus:

68% of observations (amount of time customers spend in 195℉ Coffee Company fall within 25.5 minutes and 14.62 minutes

2. For the 2nd standard deviation:

= 20.06 + 2(5.44) = 30.94

= 20.06 – 2(5.44) = 9.18

Thus:

95% of the observations fall within 30.94 minutes and 9.18 minutes

3. For the third standard deviation:

= 20.06 + 3(5.44) = 36.38

= 20.06 – 3(5.44) = 3.74

Thus:

99.7% of observations fall within 36.38 minutes and 3.74 minutes

3.5 Interquartile Range

The interquartile range, on the other hand, is a measure of the difference between the first quartile value and the third quartile value (). Just like the median, the interquartile range is considered to be a more robust means for finding the middle measurement/data value within a given dataset. The reason for this is because the interquartile range is rarely affected by outliers. Below is the formula for finding the interquartile range:

We first use MS Excel to calculate the quartiles of the given dataset by keying in the formula below:

=QUARTILE(array, quartile)

MS Excel returns the five quartiles as follows:

1. Minimum quartile (0) = 0.0121

2. First quartile (1) = 2.4661

3. Second quartile (2) = 17.2225

4. Third quartile (3) = 46.3153

5. Fourth quartile (4) = 105.0625

Hence, Interquartile range = 46.3153 – 2.4661 = 43.8492

Interquartile range = 43.85 (rounded off to two decimal places).

3.6 Evaluating Normality

This section examines if our given dataset has a normal distribution; that is, if the variables within the dataset are compatible with each and do not suffer from too many outliers, and obey the central limit theorem (). The theorem stipulates that for a dataset to have a normal distribution, the following parameters have to be fulfilled:

i) The mean, median, and mode (the measures of central tendency) have to be approximately the same.

ii) The standard deviation = 6 * Interquartile range

iii) See Table 1 below for a demonstration of the normality test values.

Table 1: Normality Test Values

Mean

20.06

Median

20.55

Standard Deviation

5.44

Interquartile range

43.85

Reading from Table 1 above:

i) The mean is 20.06 while the median is 20.55. These two values are, more or less, the same since their whole numbers is 20.

ii) The interquartile range is 43.85 while the standard deviation is 5.44. The value of interquartile range is, more or less, 8 * standard deviation.

Thus, the dataset can be said to have a normal distribution as it fulfils the aforementioned parameters.

4. The Poisson Distribution

The Poisson distribution, according to (), refers to the statistical function that estimates the number of events that occurred within a given fixed period of time. In our case, the Poisson distribution function will be useful in answering the question: What is the frequency of x? (x being the amount of time customers spend in the 195℉ Coffee Company). The formula for running the Poisson function in an MS Excel package is given by:

P(X) = X, mean, cumulative (false)

Where: p(x) = Probability of Poisson distribution of x

X = the recorded observations

See Table 2 and Figure 2 below for a demonstration of the Poisson distribution of the frequency of amounts of time customers spend at the coffee shop:

Table 2: Poisson Distribution Table

X

p(x)

24.90

0.05640

1.49

0.00000

2.66

0.00000

20.34

0.08883

0.74

0.00000

0.14

0.00000

0.10

0.00000

89.68

0.00000

13.10

0.02655

105.06

0.00000

15.13

0.05088

20.88

0.08883

7.67

0.00050

17.22

0.07527

51.70

0.00000

66.75

0.00000

50.27

0.00000

0.01

0.00000

2.40

0.00000

17.22

0.07527

0.76

0.00000

0.53

0.00000

29.38

0.01286

7.45

0.00050

84.64

0.00000

13.18

0.02655

94.09

0.00000

17.64

0.07527

34.46

0.00125

67.90

0.00000

Figure 2: Poisson Distribution Graph

5. Normal Distribution

Normal distribution refers to the plotting of a probability curve where:

i) The mean is equal to zero

ii) The mean = median = mode

iii) The standard deviation is always 1 unit from the mean

iv) The area under the curve is equal to 1

v) The plotting of the probability points generates a bell-shaped curve

The normal distribution curve presents an area under the curve which gives the probability level of the validity of a given observation (). For our case, the normal distribution curve will tell the probability level of the validity of these various amounts of time customers spend at the coffee shop. The formula for plotting a normal distribution curve in the MS Excel is as follows: = NORM.DIST(x, mean, standard_dev, cumulative). See Figure 3 below for a demonstration of the normal distribution curve for the observed amounts of time customers spend at the 195℉ Coffee shop.

Figure 3: Normal Distribution Curve of Customer Behaviour at the 195℉ Coffee Shop

6. Hypothesis Testing

As () explains, hypothesis testing refers to the usage of various statistical tools to evaluate the correctness, or lack thereof, of a given set of assumed hypotheses. The normal distribution curve is on such tool that validates the null/alternative hypotheses by calculating the test statistic of a given dataset, then comparing that test statistic with the alpha, or p-value, to determine where it lies under the curve (). If the p-value lies in the rejected region (shaded area), then the null hypothesis is null and void as it has to be rejected. However, if the p-value lies within the unshaded area, then the null hypothesis must be accepted. The computation of the hypothesis testing yearns to answer the following question: Does the duration of food preparation in 195℉ Coffee Company affect the amount of time customers spend in the shop?

i) Step 1: For our current study, the null and alternative hypotheses are:

H0 (Null Hypothesis): The duration of food preparation in the 195℉ Coffee Company affects the amount of time customers spend in the shop.

Ha (Alternative Hypothesis): The duration of food preparation in the 195℉ Coffee Company does not affect the amount of time customers spend in the shop.

ii) Step 2: Finding the test statistic:

The formula for calculating the test statistic in MS Excel is:

=Standardize(x, mean, standard_dev) = 0.000797 (test statistic)

iii) Step 3: Level of Significance = 0.05

However, since it is a two-tailed test, divide 0.05 by 2 = 0.025

iv) Compare test statistic with level of significance

0.000797 is less than 0.025; hence it lies in the unshaded shaded area which is also known as the accepted/ valid area.

Because of this, then we have to accept the null hypothesis and reject the alternative hypothesis, hence:

H0 (Null Hypothesis): The duration of food preparation in the 195℉ Coffee Company affects the amount of time customers spend in the shop - Accepted

Ha (Alternative Hypothesis): The duration of food preparation in the 195℉ Coffee Company does not affect the amount of time customers spend in the shop – Rejected

See Figure 4 below for a demonstration of the test statistic, the p-value, and the shaded/ unshaded areas under the curve.

7. Linear Relationship and Regression Analysis.

According to (), a linear relationship comprises of a mathematical equation that can be expressed graphically. Linear relationships use graphs to depict numerous statistical functions such as the regression analysis. The regression analysis is an evaluation of both the dependent and independent variables where a change in the independent variable automatically causes another change in the dependent variable (). The dependent and independent variables for this study are the customer behaviour and food preparation respectively. We shall carry out a regression analysis in a bid to answer the question: What impact does a change in food preparation have on customer behaviour in 195℉ Coffee Shop?

Step 1: First derive a linear relationship table. See Table 3 for a demonstration of this.

Time customer spends in coffee shop (mins)

Time to prepare Hot Chocolate (Mins)

25.05

2.13

21.28

1.58

18.43

1.67

24.57

1.12

20.92

1.93

19.68

1.68

20.37

1.97

10.59

1.74

16.44

1.17

9.81

1.4

23.95

1.06

15.49

1.66

17.29

1.85

24.21

1.79

12.87

2.02

28.23

1.55

27.15

1.74

19.95

1.91

18.51

1.85

24.21

2.03

20.93

1.9

19.33

1.83

25.48

1.73

22.79

1.55

29.26

1.71

16.43

1.56

10.36

1.55

15.86

1.78

14.19

2.01

28.3

1.54

Coefficients

Standard Error

t Stat

P-value

Intercept

21.51196

6.675836

3.222363

0.003218

Time to prepare Hot Chocolate (Mins)

-0.85138

3.881044

-0.21937

0.827954

Briefly explain them. Set a question based on the Coffee shop data that I send to you. Must also provide the Scatter Plot graph, the linear relationship worksheet table (make with Microsoft excel) and the Regression worksheet table (make with Microsoft Excel). Calculation and stepsare needed, you may follow the sample that I send to you as a guideline. ** You need to explain the relationship between X and Y, (indirect or direct &strong or weak relationship.)

April 06, 2023

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