Assignment 1 1430

Q. 1      Describe the formation of frequency distribution and characteristics of relative frequency distribution.

Formation of Frequency Distribution

Frequency distribution is a way to organize data into categories or intervals to see how often each category or interval occurs. Here’s how to form a frequency distribution:

1. Collect Data: Start with the raw data that needs to be analyzed.
2. Determine the Range: Find the difference between the highest and lowest values in the data set.
3. Decide the Number of Intervals (Bins): Choose how many intervals you want. The number of intervals usually depends on the size of the data set, and the goal is to make the data more interpretable. The number of intervals can be determined by Sturges’ formula: k=1+3.322log⁡nk = 1 + 3.322 \log nk=1+3.322logn, where kkk is the number of intervals and nnn is the number of observations.
4. Calculate the Interval Width: Divide the range by the number of intervals. The interval width is typically rounded to a convenient number.Interval Width=RangeNumber of Intervals\text{Interval Width} = \frac{\text{Range}}{\text{Number of Intervals}}Interval Width=Number of IntervalsRange​
5. Set the Intervals: Start from the lowest value and create intervals, ensuring they cover the entire data range without overlap.
6. Tally the Data: Count the number of data points falling into each interval.
7. Create the Frequency Distribution Table: List each interval alongside the corresponding frequency (the number of data points in that interval).

Characteristics of Relative Frequency Distribution

Relative Frequency Distribution provides a way to understand the proportion of the total number of data points that fall within each interval, relative to the entire data set. Here are its key characteristics:

1. Proportional Representation: Relative frequency is calculated by dividing the frequency of each interval by the total number of observations. It shows the proportion or percentage of the total data that falls within each interval.Relative Frequency=Frequency of IntervalTotal Number of Observations\text{Relative Frequency} = \frac{\text{Frequency of Interval}}{\text{Total Number of Observations}}Relative Frequency=Total Number of ObservationsFrequency of Interval​
2. Sum Equals 1: The sum of all relative frequencies in a distribution is equal to 1 (or 100% when expressed as a percentage), which makes it easy to compare different datasets of varying sizes.
3. Interpretability: It provides a clearer interpretation of how data is distributed across intervals, especially when the dataset is large, as it allows for comparison of the relative importance or likelihood of different outcomes.
4. Useful for Probability: Relative frequency distribution is closely related to probability distribution, making it useful in probability theory and statistics, as it can be used to estimate the probability of different outcomes.
5. Comparison Across Groups: Relative frequency distribution is particularly helpful when comparing different groups or samples because it accounts for differences in sample size.
6. Visualization: Like absolute frequency distribution, relative frequency distribution can be visualized using bar charts or histograms, but the y-axis represents proportions or percentages instead of raw counts.

Q.2   Transmission ABC stores recorded the number of service tickets submitted by 50stores last month as follows:                                                                                                              (10+10)

823 648 321 634 752 669 427 555 904 586 722 360 468 847 641 217 588 349 308 766 114 163 150 718 687 763 607 441 305 662 227 624 791 960 334 163 550 842 860 413 439 981 416 115 810 957 919 846 365 924

• Arrange the data using the Data Array and the Frequency Distribution.
• Which arrangement you prefer? Explain.

(a) Data Arrangement

1. Data Array (Ascending Order):

To arrange the data in a data array, we sort the numbers in ascending order:

114, 115, 150, 163, 163, 217, 227, 305, 308, 321, 334, 349, 360, 365, 413, 416, 427, 439, 441, 468, 550, 555, 586, 588, 607, 624, 634, 641, 648, 662, 669, 687, 718, 722, 752, 763, 766, 791, 810, 823, 842, 846, 847, 860, 904, 919, 924, 957, 960, 981.

2. Frequency Distribution:

To create a frequency distribution, we categorize the data into intervals (bins) and count how many data points fall into each interval. Here, we’ll choose a bin width of 100:

(b) Preferred Arrangement and Explanation:

Frequency Distribution is generally preferred over a simple data array, especially when analyzing a large dataset. Here’s why:

1. Simplification: Frequency distribution summarizes the data by grouping it into intervals, making it easier to identify trends and patterns without having to examine each individual data point.
2. Visualization: It allows for easy visualization through histograms or bar charts, making it simpler to understand the distribution of data.
3. Analysis: Frequency distributions are useful for statistical analysis, such as calculating the mode, median, or identifying skewness in the data, which is more challenging when looking at a raw data array.

In contrast, a data array is useful when the exact value of each data point is needed, but it can be overwhelming and difficult to interpret when dealing with larger datasets like this one.

Q.3   (a)     Describe different measures of central tendency.                              (10+10)

         (b)     Here are the ages of forty-eightmembers of a country service program:

                  83     51     66     61     82     65     54     56     92     60     65     87

                  68     64     51     70     75     66     74     68     44     55     78     69

                  98     67     82     77     79     62     38     88     76     99     84     47

                  60     42     66     74     91     71     83     80     68     65     51     56                         Obtain (i) median, (ii) mode (iii) range and coefficient of range

(a) Measures of Central Tendency

1. Mean: The mean is the average of a set of numbers. It is calculated by summing all the values in the dataset and dividing by the number of values. Mean=∑Xi​​/n

Where Xi​ represents each value in the dataset and n is the total number of values.

2. Median: The median is the middle value in a dataset when the numbers are arranged in ascending or descending order. If there is an odd number of observations, the median is the middle number. If there is an even number, the median is the average of the two middle numbers.

• For an odd number of data points:

Median=X^_(n+1/2​)​

• For an even number of data points:

Median=X_(n​/2)​+X(_n/2​+1)​​ /2

3. Mode: The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode if all values occur with the same frequency.

4. Midrange: The midrange is the average of the maximum and minimum values in the dataset. It provides a simple measure of central tendency but is highly sensitive to outliers.

Midrange=Max+Min​/2

5. Geometric Mean: The geometric mean is calculated by multiplying all the numbers together and then taking the nnnth root (where nnn is the total number of values). It’s useful when dealing with data that is multiplicative in nature, such as growth rates.

Geometric Mean=(i=1∏n​Xi​)^1/n​

6. Harmonic Mean: The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data values. It is used when the data involves rates or ratios

Harmonic Mean=n/∑1/Xi​

(b) Calculations for the Given Data

Dataset: 83, 51, 66, 61, 82, 65, 54, 56, 92, 60, 65, 87, 68, 64, 51, 70, 75, 66, 74, 68, 44, 55, 78, 69, 98, 67, 82, 77, 79, 62, 38, 88, 76, 99, 84, 47, 60, 42, 66, 74, 91, 71, 83, 80, 68, 65, 51, 56.

Step 1: Arrange the Data in Ascending Order 38, 42, 44, 47, 51, 51, 51, 54, 55, 56, 56, 60, 60, 61, 62, 64, 65, 65, 65, 66, 66, 66, 67, 68, 68, 68, 69, 70, 71, 74, 74, 75, 76, 77, 78, 79, 80, 82, 82, 83, 83, 84, 87, 88, 91, 92, 98, 99.

(i) Median:

• Since there are 48 data points (even number), the median is the average of the 24th and 25th values.
• The 24th and 25th values are both 68.
• Median=68+68​/2 =68

(ii) Mode:

• The mode is the value that appears most frequently.
• Here, 51, 65, 66, and 68 each appear 3 times, making the dataset multimodal.
• Mode=51,65,66,68

(iii) Range and Coefficient of Range:

• Range:
• Range=Max−Min=99−38=61
• Coefficient of Range:
• Coefficient of Range=Max+Min/Max−Min​=99+38/99−38​=61/137​≈0.445

So, the median is 68, the mode values are 51, 65, 66, and 68, the range is 61, and the coefficient of range is approximately 0.445.

Q. 4   From the following data calculate (i) mean, (ii) geometric mean, (iii) standard deviation and (iv) coefficient of Variation (CV)                                                                                    (5+5+5+5)

Q.5    (a)     Discuss the basic concepts in hypothesis-testing procedure.               (10+10)

         (b)     The average commission charged by full-service brokerage firms on a sale of common stock is $144 and standard deviation is $52. Joel Freelander has taken a random sample of 121 trades by his clients and determined that they paid an average commission of $151. At a 0.10 significance level, can Joel conclude that his clients commissions are higher than the industry average.

Part (a): Basic Concepts in Hypothesis-Testing Procedure

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. The basic concepts involved in the hypothesis-testing procedure include:

1. Null Hypothesis (H₀):
   • The null hypothesis is a statement of no effect, no difference, or no change. It is the hypothesis that the researcher seeks to test, usually representing a default or status quo condition.
   • Example: “The average commission charged by Joel’s clients is equal to the industry average of $144.”
2. Alternative Hypothesis (H₁ or Ha):
   • The alternative hypothesis is a statement that contradicts the null hypothesis. It represents the effect or difference that the researcher expects or hopes to find.
   • Example: “The average commission charged by Joel’s clients is higher than the industry average of $144.”
3. Significance Level (α):
   • The significance level is the probability of rejecting the null hypothesis when it is actually true. It is the threshold for determining whether the observed data is sufficiently unusual to reject the null hypothesis.
   • Common significance levels are 0.05, 0.01, and 0.10. In this problem, the significance level is 0.10.
4. Test Statistic:
   • A test statistic is a standardized value calculated from sample data, which is then compared against a critical value to determine whether to reject the null hypothesis.
   • The choice of test statistic depends on the type of data and the specific hypothesis test being used. Common test statistics include the z-statistic, t-statistic, and chi-square statistic.
5. P-Value:
   • The p-value is the probability of obtaining a test statistic at least as extreme as the one calculated from the sample data, assuming that the null hypothesis is true. It is used to determine the statistical significance of the results.
   • If the p-value is less than or equal to the significance level (α), the null hypothesis is rejected.
6. Critical Value:
   • The critical value is the value that separates the rejection region from the non-rejection region in a hypothesis test. It is determined by the significance level and the type of test (one-tailed or two-tailed).
   • If the test statistic exceeds the critical value, the null hypothesis is rejected.
7. Decision Rule:
   • The decision rule outlines the criteria for rejecting or failing to reject the null hypothesis. It is based on comparing the test statistic to the critical value or comparing the p-value to the significance level.
   • Example: “Reject the null hypothesis if the z-statistic is greater than the critical value.”
8. Type I and Type II Errors:
   • Type I Error (α): Occurs when the null hypothesis is rejected when it is true. The probability of making a Type I error is the significance level.
   • Type II Error (β): Occurs when the null hypothesis is not rejected when it is false. The probability of making a Type II error is denoted by β, and its complement (1 – β) is the power of the test.
9. One-Tailed vs. Two-Tailed Tests:
   • One-Tailed Test: Tests for a difference in a specific direction (e.g., greater than or less than a certain value).
   • Two-Tailed Test: Tests for a difference in either direction (e.g., different from a certain value, either greater or less).
10. Conclusion:
    • Based on the test statistic and p-value, a conclusion is drawn about whether to reject or fail to reject the null hypothesis. This conclusion should be interpreted in the context of the research question.