Scenario Recall For This Week's Discussion
Scenariorecall That For This Weeks Discussion You Considered Data Re
Scenario: Recall that for this week’s Discussion you considered data related to opening or attracting a new restaurant. Now consider that you ask 20 participants to estimate how many times a month they go out to dinner and you receive these responses: 1, 2, 5, 8, 2, 4, 8, 4, 2, 3, 6, 8, 7, 5, 8, 4, 0, 7, 6, and 18. Assignment: To complete this Assignment, submit by Day 7 calculations of the following measures of central tendency and variability using the data set provided. Include an explanation of how you calculated each measure and what information each measure gives you about the dining behavior of the sample. Finally, create a data file in SPSS and run analyses to find the mean and standard deviation.
Note: Your hand-calculated mean and standard deviation will differ somewhat from the calculations in SPSS due to rounding. Hand-calculated mean: SPSS mean: Median: Mode: Range: Deviation of the highest score from the mean: Hand-calculated standard deviation (Please also state the hand-calculated values for ΣX2 and (ΣX)2): SPSS standard deviation: Explain how the standard deviation (SD) and the deviation of a single score differ in the information they provide. Explain how each measure (mean, median, mode, deviation of the highest score from the mean, and standard deviation) would change if the score of 18 was eliminated from the data set. Explain the type of distribution (positive skew, negative skew, bimodal distribution, or normal distribution) your data create. Explain how you know the type of distribution and what the data tells you about your sample. Submit three documents for grading: your text (Word) document with your answers and explanations to the application questions, your SPSS Data file, and your SPSS Output file. Resources: Readings Heiman, G. (2015). Behavioral sciences STAT 2 (2nd ed). Stamford, CT: Cengage. Review Section 2-3 “Types of Frequency Distributions” (pp.25-28) Chapter 3, “Summarizing Scores with Measures of Central Tendency” (pp.36-49) Chapter 4, “Summarizing Scores with Measure of Variability” (pp.52-65) Chapter 1 Review Card (p. 1.4) Chapter 3 Review Card (p. 3.4) Using the attached dataset, and the Frequency command, calculate the standard descriptive measures (mean, median, mode, standard deviation, variance, and range) as well as kurtosis and skew for all three hygiene variables. Show all work and output. What does the output tell us? Comment on sample size, measures of central tendency and dispersion as well as kurtosis and skewness. Also, calculate z scores for skewness and kurtosis. Is the assumption of normality met for these three variables?
Paper For Above instruction
The analysis of data pertaining to consumer behavior, such as the frequency of dining out, can provide valuable insights into patterns and tendencies that influence business strategies, particularly in the hospitality industry. This paper will perform a comprehensive statistical assessment of a sample dataset where 20 participants provided their estimated number of times they dine out each month. The focus will be on calculating key measures of central tendency and variability, interpreting these metrics, and understanding the distributional characteristics of the data, supplemented by SPSS analyses.
Calculating Descriptive Statistics
The dataset, comprising the responses: 1, 2, 5, 8, 2, 4, 8, 4, 2, 3, 6, 8, 7, 5, 8, 4, 0, 7, 6, and 18, serves as the basis for computing measures such as mean, median, mode, range, standard deviation, and deviation from the mean. The hand-calculated mean involves summing all responses and dividing by the total number of observations:
\[
\text{Sum of responses} (\Sigma X) = 1+2+5+8+2+4+8+4+2+3+6+8+7+5+8+4+0+7+6+18= 124
\]
\[
\text{Mean} = \frac{\Sigma X}{N} = \frac{124}{20} = 6.2
\]
The median, which is the middle value in an ordered dataset, is identified by first ordering the responses from smallest to largest:
0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 18
With 20 observations, the median is the average of the 10th and 11th values:
\[
\text{Median} = \frac{5 + 5}{2} = 5
\]
The mode, the most frequently occurring response, in this case is 8, which appears four times, making it the modal value. The range is calculated by subtracting the smallest from the largest:
\[
\text{Range} = 18 - 0 = 18
\]
The deviation of the highest score from the mean is:
\[
18 - 6.2 = 11.8
\]
Variability Measures
To compute the variance and standard deviation, the sum of squared deviations (ΣX²) is essential. Calculating each response's deviation from the mean, squaring it, and summing provides ΣX²:
Responses and their deviations:
| Response | Deviation from mean | Squared deviation |
|------------|---------------------|-------------------|
| 0 | -6.2 | 38.44 |
| 1 | -5.2 | 27.04 |
| 2 | -4.2 | 17.64 |
| 2 | -4.2 | 17.64 |
| 2 | -4.2 | 17.64 |
| 3 | -3.2 | 10.24 |
| 4 | -2.2 | 4.84 |
| 4 | -2.2 | 4.84 |
| 4 | -2.2 | 4.84 |
| 5 | -1.2 | 1.44 |
| 5 | -1.2 | 1.44 |
| 6 | -0.2 | 0.04 |
| 6 | -0.2 | 0.04 |
| 7 | 0.8 | 0.64 |
| 7 | 0.8 | 0.64 |
| 8 | 1.8 | 3.24 |
| 8 | 1.8 | 3.24 |
| 8 | 1.8 | 3.24 |
| 8 | 1.8 | 3.24 |
| 18 | 11.8 | 139.24 |
Summing these squared deviations gives:
\[
\Sigma X^2 = 38.44 + 27.04 + (3 \times 17.64) + 10.24 + (3 \times 4.84) + 1.44 + 1.44 + 0.04 + 0.04 + 0.64 + 0.64 + (4 \times 3.24) + 139.24 = 386.4
\]
The variance (s²) is:
\[
s^2 = \frac{\Sigma (X_i - \bar{X})^2}{N - 1} = \frac{386.4}{19} \approx 20.34
\]
Standard deviation (s):
\[
s = \sqrt{20.34} \approx 4.51
\]
These are the hand-calculated measures, noting that in SPSS, rounding differences will occur.
Comparison of Standard Deviation and Deviation of a Single Score
The standard deviation summarizes the average amount of variability in the data set and is a measure of how spread out the responses are around the mean. Conversely, the deviation of a single score (e.g., 18 from the mean of 6.2) shows how far that individual response is from the central tendency but offers no information about overall variability.
Effect of Eliminating the Highest Score (18)
Removing the outlying high response of 18 would decrease the mean, bringing it closer to the majority of responses, likely around 4-6. The median might remain stable, as it is less affected by outliers. The mode would likely stay at 8. The range would decrease substantially from 18 to 8, since the maximum would be 8. The variance and standard deviation would decrease, indicating less spread in the data, leading to a distribution more concentrated near the central value.
Distribution Characteristics
Plotting or analyzing skewness and kurtosis informs about the distribution's shape. Logistic examination suggests a right-skewed distribution (positive skew) primarily due to the outlier of 18, which pulls the mean upward relative to the median. The distribution's skewness and kurtosis coefficients, along with their z-scores, can statistically test this, where significant z-scores indicate departure from normality.
Conclusion
Overall, these descriptive measures provide a detailed picture of dining behaviors. The data suggest a positively skewed distribution, with most participants dining out around 2-8 times per month, but a few individuals dining out much more frequently, skewing the data.
References
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