MAT 274 Descriptive Statistics Project

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MAT 274 Descriptive Statistics Project

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MAT 274 Descriptive Statistics Project

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Course Code: MAT274
University: Grand Canyon University

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Country: United States

Questions:

Choose one of the attached (added as a attached file) data sets and analyze using the techniques discussed in class up to this point. This includes the following:
 
1. Find the appropriate measure of center. Discuss why the chosen measure is most appropriate. Why did you decide against other possible measures of center?2. Find the appropriate measure of variation. The measure of variation chosen here should match the measure of center chosen in Part 1.3. Find the graph(s) needed to appropriately describe the data. These may be done by hand and inserted into the Word document.4. Define a random variable (X) so that your chosen data set represents values of X.5. Is your chosen random variable discrete or continuous? Explain how you know.6. Would the Normal or Binomial distribution be a good fit for the underlying sample distribution of X? If one of them is a good fit, state how you would approximate the distribution parameters.7. Calculate the probability that a flight will depart early or on-time.8. Calculate the probability that a flight will arrive late.9. Calculate the probability that a flight departs late or arrives early.10. Assume now that the random variable X=Arrival Time is exactly normally distributed with mean m= -2.5 and standard deviation s= 23. Compute the probability of a flight arriving late based on this new information. Does this contradict your answer from Part 8?

Answers:
Description of Dataset
The sample dataset provided was provided from the department of transportation and statistics. It represents “reporting carrier on-time performance.” It has got four variables; day_of_month and day_of_week are discrete variables that indicates the day of the month and the day of the week when the data was collected respectively. On the other hand, dep_delay and arr_delay are continuous variables indicating the difference that exists in minutes between scheduled and actual departure and arrival times respectively. Early arrival or departure is denoted by a negative sign. From the data, it is evident that in the year 2015, five hundred observations on arrival and departure delay were taken on the thirteenth day of a given month and the second day of a given week in the month. The is normally distributed with an average departure delay time of -5.296 and an average arrival time was -3.138.
The dataset provided is to be used for carrying out statistical analysis such as determining the measure of central tendency and variation, creating a plot that best describes the data using an appropriate graph and use an appropriate probability distribution to estimate parameters. Additionally, the dataset will be used as an aid to examine statistical viability sources, the significance of randomness in statistical inference and to analyze the difference that exists between empirical and theoretical probability and they impact they have on probability calculations. 
The table for descriptive statistics as developed from excel is as shown below and is used to answer question 1 and 2.

To find the appropriate measure of center and explain why it is most appropriate.

The commonly used measures of centers are the mean and the median. However, in this case, the two variables under consideration; dep_delay and arr_delay are somewhat uniformly distributed and there are no observable outliers hence the most appropriate measure of center is the mean. The mean for dep_delay is -5.296 while the mean for arr_delay is -3.138.

To find the most appropriate measure of variation for the data.

The commonly used measure of variation is the standard deviation, the variance and the range. However, due to the same reason as in question 1, the most appropriate measure of variation in this case for the variables is the standard deviation. The standard deviation for dep_delay is 9.21 while that for arr_delay is 9.66.

To find the graph needed to most appropriately describe the sample data provided.

The purpose of representing the data provided graphically would be to determine the relationship that exist between the data, that is how the arrival delay and departure delay are related. In that case we use a scatter plot which would indicate the correlation between the variables using dots indicating the data points. A trend line is added to draw a valid conclusion on the relationship (Freund, 2014).

Defining a random variable (X) to represents value of the variables in the data.

Let X be a given random variable that will give a description of the departure and arrival time of the flights.

Explaining whether the random variable X is discrete or continuous.

The chosen random variable X is a continuous variable. This is because despite the time being in minutes, there could be a possibility of having countable number of seconds in between two consecutive minutes for departure and arrival.

Explanation of what between normal and binomial distribution is a good fit for the sample of X and how to describe how approximate the parameter distributions.

The normal distribution would be a good fit for the underlying sample because firstly the sample data is continuous and secondly because going by the central limit theorem, for a large sample size random variable would follow a normal distribution. Distribution parameters in a normal distribution are approximated from the sample mean (Foster, Barkus & Yavorsky, 2006).
Probability that a flight departs early or exactly on –time.
The probability that it will depart early or on-time is given by number of flights that arrive without delay divided by the number total number of flights as below:
Probability of late arrival for the plain.
The probability that the flight will arrive late is given by the total count of late arrivals divided by the total count of all arrivals.
Probability whether a flight departs late or the flight arrives early. 
The probability for late departure or early arrival is given by the adding the probability of late departure and the probability of early arrivals
Computation the probability of a flight arriving late based on the new information and determining whether the answer contradict the answer obtained in part 8
Since the average and the standard deviation are given, the probability from a population mean is determined using the formula:
In excel, after determining the z-score the function NORMSDIST(Z) is used to determine the probability (Linoff, 2008).
The probability of a flight arriving late is 0.2675 and Yes, the answer in this case contradicts the answer of question eight mostly because of poor approximation the population mean and the standard deviation.
References
Foster, J. J., Barkus, E., & Yavorsky, C. (2006). Understanding and using advanced statistics (2nd ed.). London: SAGE.
Freund, J. E. (2014). Modern elementary statistics (12th ed.). Boston: Pearson.
Linoff, G. (2008). Data analysis using SQL and Excel. Indianapolis, Ind.: Wiley Pub.

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