Explain A: The Normal Distribution And B: The Empirical

Explain A The Normal Distribution And B The Empirical

Instructions explain (a) the normal distribution and (b) the Empirical Rule. Assume that the distribution of daily sales for laptops are normally distributed, with a mean of 4760 units and a standard deviation of 510. Answer the following questions: a. Between what two values would you expect 95% of the days' sales to fall? b. What percentage of days' sales would be expected to fall between 1600 and 3400? c. What percentage of days' sales would be expected to be more than 3400? d. What percentage of days' sales would be expected to be more than 5500 units? e. If you took a sample of 75 days from the previous year and found that 50 of those days had sales that ranged from 3875 to 5325 units, would that indicate a potential problem with the data? Or would that be in line with your expectations? Explain your rationale for your answer. In addition, you should create a "cheat sheet" for selecting an appropriate statistical test. Your "cheat sheet" should include information on when you would use the following tests: Single sample t test, Independent samples t test, Paired samples t test, One-way ANOVA, Factorial ANOVA, ANCOVA, Chi-Square Goodness of Fit, Chi-Square Test of Independence, Pearson Correlation Coefficient, Simple Linear Regression, Multiple Linear Regression, Spearman rho correlation.

Paper For Above instruction

The assessment of data distribution often relies on understanding the properties of the normal distribution and the application of the Empirical Rule. In this context, we examine the distribution of daily laptop sales, which, based on the given parameters, follows a normal distribution with a mean of 4760 units and a standard deviation of 510 units. This statistical framework allows for probabilistic predictions about sales figures and aids in making informed business decisions.

The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve. It is symmetrical around the mean, which represents the central tendency of the data. The properties of the normal distribution include its symmetry, the empirical rule for describing data spread, and the fact that approximately 68% of the data falls within one standard deviation, about 95% within two standard deviations, and about 99.7% within three standard deviations from the mean.

Harking back to the example of daily laptop sales, this distribution assumes that sales figures are spread in a pattern that clusters around the average, with fewer days experiencing very high or very low sales. The normal distribution is widely used in business analytics because many natural phenomena, including sales data, tend to approximate this distribution under certain conditions.

The Empirical Rule

The Empirical Rule provides a quick way to estimate the percentage of data within specific ranges of a normal distribution based on standard deviations from the mean. Specifically:

• Approximately 68% of data falls within one standard deviation (mean ± 1σ)
• Approximately 95% of data falls within two standard deviations (mean ± 2σ)
• Approximately 99.7% of data falls within three standard deviations (mean ± 3σ)

Using the given distribution parameters, we can calculate specific bounds for the sales figures. For question (a), to find the range encompassing 95% of the sales, two standard deviations from the mean are considered:

Mean = 4760 units, Standard deviation = 510 units

Calculations:

• Lower bound: 4760 - 2 × 510 = 4760 - 1020 = 3740 units
• Upper bound: 4760 + 2 × 510 = 4760 + 1020 = 5780 units

Thus, 95% of the daily sales are expected to fall between 3740 and 5780 units.

For question (b), the probability that sales fall between 1600 and 3400 units requires calculating the appropriate Z-scores and using standard normal distribution tables or software to find the corresponding percentages. Similarly, questions (c) and (d) are addressed by computing the probabilities associated with sales greater than 3400 units and 5500 units, respectively.

Finally, question (e) relates to examining whether the sample data aligns with expectations based on the normal distribution. The observed range (3875 to 5325 units) for 50 days out of 75 is consistent with the expected 95% range, suggesting the data does not indicate a problem. This aligns with the properties of the normal distribution, which predict that about 95% of observations should fall within two standard deviations of the mean.

Statistical Test Selection Cheat Sheet

Choosing the appropriate statistical test depends on the research question, the data type, and the design of the study. The following guidelines can assist in selecting the proper test:

• Single sample t test: Used to compare the mean of a single sample to a known value or hypothesized population mean.
• Independent samples t test: Compares the means of two independent groups to see if they differ significantly.
• Paired samples t test: Compares means from the same group at two different times or under two different conditions.
• One-way ANOVA: Tests for differences among the means of three or more independent groups based on one factor.
• Factorial ANOVA: Examines the effect of two or more factors on a dependent variable, including interaction effects.
• ANCOVA: Compares one or more independent groups while controlling for covariates.
• Chi-Square Goodness of Fit: Tests whether observed categorical data fit an expected distribution.
• Chi-Square Test of Independence: Assesses whether two categorical variables are independent.
• Pearson Correlation Coefficient: Measures the strength and direction of the linear relationship between two continuous variables.
• Simple Linear Regression: Models the relationship between one independent variable and a dependent variable.
• Multiple Linear Regression: Extends simple linear regression to include multiple independent variables.
• Spearman rho correlation: A non-parametric measure of the rank correlation between two variables, used when data are not normally distributed.

References

• Bolstad, W. M. (2016). Introduction to Bayesian Statistics. Wiley.
• Daniel, W. W. (2018). Biostatistics: A Foundation for Analysis in the Health Sciences. Wiley.
• Field, A. (2013). Discovering Statistics Using SPSS. Sage Publications.
• Loftus, G. R. (2018). Regression Modeling Strategies. Chapman and Hall/CRC.
• McDonald, J. H. (2014). Handbook of Biological Statistics. Sparky House Publishing.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.
• Field, A. (2018). Discovering Statistics Using R. Sage Publications.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
• Sheskin, D. J. (2011). Handbook of Parametric and Nonparametric Statistical Procedures. Chapman and Hall/CRC.
• Yen, W. M. (2014). Statistical Methods in Educational Research. Routledge.