Discuss the role of statistics in research.

Answer:

Compute Spearman’s Rank Correlation for the following data:

Answer:

1. Assign Ranks: Rank the values of $X$$X$XX$X$ and $Y$$Y$YY$Y$ separately, from smallest to largest (1 for the smallest value). If there are ties, assign the average rank.
2. Calculate Rank Differences: Find the difference between the ranks of $X$$X$XX$X$ and $Y$$Y$YY$Y$ for each pair ($d_i={\text{Rank}}_X-{\text{Rank}}_Y$$d_i={\text{Rank}}_X-{\text{Rank}}_Y$d_(i)=”Rank”_(X)-“Rank”_(Y)d_i = \text{Rank}_X – \text{Rank}_Y$d_i={\text{Rank}}_X-{\text{Rank}}_Y$).
3. Square the Differences: Compute $d_i^2$$d_i^2$d_(i)^(2)d_i^2$d_i^2$ for each pair.
4. Sum the Squared Differences: Calculate the sum of $d_i^2$$d_i^2$d_(i)^(2)d_i^2$d_i^2$.
5. Apply the Spearman Formula: Use the formula:$$\rho =1-\frac{6\sum d_i^2}{n(n^2-1)}$$$$\rho =1-\frac{6\sum d_i^2}{n(n^2-1)}$$rho=1-(6sumd_(i)^(2))/(n(n^(2)-1))\rho = 1 – \frac{6 \sum d_i^2}{n(n^2 – 1)}$$\rho =1-\frac{6\sum d_i^2}{n(n^2-1)}$$where $n$$n$nn$n$ is the number of data pairs.

Step 1: Assign Ranks

• For $X$$X$XX$X$: The values in ascending order are 16 (1), 17 (2), 18 (3), 20 (3), 25 (4), 27 (5), 28 (6), 29 (7), 40 (9), 60 (10). Note that 18 and 20 are tied for the 3rd and 4th positions, so they both get the average rank $(3+4)/2=3.5$$(3+4)/2=3.5$(3+4)//2=3.5(3 + 4)/2 = 3.5$(3+4)/2=3.5$. However, since they are not exact ties in the standard ranking (we assign ranks sequentially), we adjust accordingly.
• For $Y$$Y$YY$Y$: The values in ascending order are 15 (1), 16 (2), 20 (3), 24 (4), 25 (5), 26 (6), 27 (7), 28 (8), 55 (9), 35 (10).
• The ranks for $X=18$$X=18$X=18X = 18$X=18$ and $X=20$$X=20$X=20X = 20$X=20$ are adjusted as 3 and 3, respectively, in standard ranking practice for simplicity, as they are distinct values.

Step 2: Calculate Rank Differences ($d_i$$d_i$d_(i)d_i$d_i$)

Step 3: Square the Differences ($d_i^2$$d_i^2$d_(i)^(2)d_i^2$d_i^2$)

Step 4: Sum the Squared Differences

Step 5: Apply the Spearman Formula

Final Answer

Explain Descriptive and Inferential statistics.

Answer:

Explain the key components of tabulation.

Answer:

1. Title: The title of the table clearly defines the subject or purpose of the table, indicating the data being presented.
2. Rows: Rows represent individual items or categories of data. Each row typically corresponds to a specific observation, group, or value.
3. Columns: Columns represent different variables or characteristics of the data. They categorize the data according to specific attributes or measurement types.
4. Cells: The intersection of rows and columns forms cells, which contain the actual data values or frequencies. These cells present the information being summarized in the table.
5. Headings: Each row and column is labeled with headings to indicate the specific data being presented. These headings help in identifying and understanding the data in the table.
6. Subtotals and Totals: Subtotals are used to summarize data for specific categories, and the total row or column presents the overall aggregate.
7. Source Notes: These provide information on the origin of the data or any other necessary clarifications for the table’s interpretation.

Describe the different types of frequency distribution.

Answer:

1. Univariate Frequency Distribution: This type deals with a single variable. It counts the occurrences of each unique value in the dataset. For example, counting how many times each score appears in a set of test results.
2. Bivariate Frequency Distribution: This type involves two variables, showing the relationship between them. It counts the occurrences of paired values, often displayed in a contingency table.
3. Cumulative Frequency Distribution: This shows the accumulation of frequencies up to a certain value. It helps in understanding the proportion of data points that fall below a specific value or range.
4. Relative Frequency Distribution: Instead of absolute counts, this distribution shows the proportion of data points that fall into each class interval. It is calculated by dividing the frequency of each class by the total number of observations.
5. Grouped Frequency Distribution: When data has many unique values, it is grouped into intervals or classes to simplify the presentation. This is helpful for large datasets where individual values may not be meaningful on their own.

Compute the mean, median and mode for the following data:

10, 12, 8, 9, 10, 14, 10, 15, 8, 8

Answer:

Step 1: Organize the Data

Step 2: Compute the Mean

Step 3: Compute the Median

Step 4: Compute the Mode

• $8$$8$88$8$: 3 times
• $9$$9$99$9$: 1 time
• $10$$10$1010$10$: 3 times
• $12$$12$1212$12$: 1 time
• $14$$14$1414$14$: 1 time
• $15$$15$1515$15$: 1 time

Final Answer

• Mean: 10.4
• Median: 10
• Mode: 8 and 10 (bimodal)

Compute quartile deviation for the following data:

40, 43, 44, 48, 52, 53, 57, 58, 60, 62

Answer:

Step 1: Understand Quartile Deviation

• $Q_1$$Q_1$Q_(1)Q_1$Q_1$ is the first quartile (25th percentile).
• $Q_3$$Q_3$Q_(3)Q_3$Q_3$ is the third quartile (75th percentile).

Step 2: Organize the Data

Step 3: Find $Q_1$$Q_1$Q_(1)Q_1$Q_1$ (First Quartile)

• 2nd value = 43
• 3rd value = 44

Step 4: Find $Q_3$$Q_3$Q_(3)Q_3$Q_3$ (Third Quartile)

• 8th value = 58
• 9th value = 60

Step 5: Calculate Quartile Deviation

Final Answer

Explain skewness and kurtosis.

Answer:

• Positive skew: The right tail is longer, and the mean is greater than the median.
• Negative skew: The left tail is longer, and the mean is less than the median.

• Leptokurtic: Distributions with heavy tails and sharp peaks, indicating more extreme values (outliers).
• Platykurtic: Distributions with light tails and a flatter peak, indicating fewer outliers.
• Mesokurtic: A normal distribution with kurtosis equal to 3, representing a moderate peak.