State and explain seven (7) assumptions that are made when using parametric statistics to test a hypothesis.

1. State and explain seven (7) assumptions that are made when using parametric statistics to test a hypothesis. • Normality: The data for each group should be approximately normally distributed. This assumption is crucial for the validity of many parametric tests. • Homogeneity of Variance: The variance of the dependent variable should be equal across all groups or levels of the independent variable. This is often checked using Levene's test. • Independence of Observations: Each observation or data point must be independent of every other observation. This means that the measurement of one subject does not influence the measurement of another. • Interval or Ratio Data: The dependent variable must be measured on an interval or ratio scale, meaning it has meaningful numerical differences and, for ratio scales, a true zero point. • Absence of Outliers: Parametric tests are sensitive to extreme values (outliers), which can distort the results. Outliers should be identified and addressed appropriately. • Linearity: For tests involving relationships between two continuous variables (e.g., regression), the relationship between the variables should be linear. • Random Sampling: The samples used in the study should be randomly selected from the population to ensure they are representative and to allow for generalization of results. 2. Explain the following terms in details. a) Population. A population is the entire group of individuals, objects, or events that share a common characteristic and are of interest to a researcher. For example, all students enrolled in a particular university. b) Confounding variables. Confounding variables are extraneous variables that are related to both the independent and dependent variables, potentially distorting the true relationship between them. For example, in a study on coffee and heart disease, smoking could be a confounding variable. c) Non-probability sampling. Non-probability sampling is a sampling technique where the selection of participants is not based on random chance, meaning some members of the population have no chance of being selected. Examples include convenience sampling or snowball sampling. d) Symmetry. In statistics, symmetry refers to the shape of a distribution where the left and right sides of the distribution are mirror images of each other around the central point (mean). A normal distribution is a classic example of a symmetric distribution. e) Standard deviation. Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. 3. (a) Obtain the standard deviation of the following data; 15, 10, 16, 15, 17, 20, 10, 19, 18, and 20. Step 1: List the data and calculate the number of data points (n). The data points are 15, 10, 16, 15, 17, 20, 10, 19, 18, 20. The number of data points is n = 10. Step 2: Calculate the mean (x). x = ( x)/(n) = (15+10+16+15+17+20+10+19+18+20)/(10) = (160)/(10) = 16 Step 3: Calculate the squared difference from the mean for each data point and sum them. (x - x)^2 = (15-16)^2 + (10-16)^2 + (16-16)^2 + (15-16)^2 + (17-16)^2 + (20-16)^2 + (10-16)^2 + (19-16)^2 + (18-16)^2 + (20-16)^2 = (-1)^2 + (-6)^2 + (0)^2 + (-1)^2 + (1)^2 + (4)^2 + (-6)^2 + (3)^2 + (2)^2 + (4)^2 = 1 + 36 + 0 + 1 + 1 + 16 + 36 + 9 + 4 + 16 = 120 Step 4: Calculate the variance (^2) for a population. ^2 = (x - x)^2n = (120)/(10) = 12 Step 5: Calculate the standard deviation (). = sqrt(^2) = sqrt(12) ≈ 3.464 The standard deviation is 3.464. 3. (b) Explain the following terms with examples: i. The mode. The mode is the value that appears most frequently in a data set. A data set can have one mode (unimodal), multiple modes (multimodal), or no mode. For example, in the data set \2, 3, 3, 4, 5, 5, 5, 6\, the mode is 5. ii. The mean. The mean (or arithmetic mean) is the average of a set of numbers, calculated by summing all the values and dividing by the count of values. For example, for the data set \2, 4, 6, 8\, the mean is (2+4+6+8)/(4) = (20)/(4) = 5. iii. The median. The median is the middle value in a data set when the values are arranged in ascending or descending order. If there is an even number of data points, the median is the average of the two middle values. For example, in the data set \1, 3, 5, 7, 9\, the median is 5. In the data set \1, 3, 5, 7\, the median is (3+5)/(2) = 4. 4. (a) Find the mean number of coins. Step 1: Determine the midpoints (x) for each class interval. • 0-4: Midpoint = (0+4)/(2) = 2 • 5-7: Midpoint = (5+7)/(2) = 6 • 8-10: Midpoint = (8+10)/(2) = 9 • 11-12: Midpoint = (11+12)/(2) = 11.5 Step 2: Create a table with midpoints (x), frequencies (f), and the product f × x. | No. of coin (Class) | Midpoint (x) | No. of people (f) | f × x | | :------------------ | :------------- | :------------------ | :----------- | | 0-4 | 2 | 6 | 12 | | 5-7 | 6 | 8 | 48 | | 8-10 | 9 | 8 | 72 | | 11-12 | 11.5 | 8 | 92 | Step 3: Calculate the sum of f × x and the sum of frequencies ( f). (f × x) = 12 + 48 + 72 + 92 = 224 f = 6 + 8 + 8 + 8 = 30 Step 4: Calculate the mean (x). x = ( (f × x))/( f) = (224)/(30) ≈ 7.467 The mean number of coins is 7.467. 4. (b) Define quartiles, deciles and parametric. Quartiles are values that divide a data set into four equal parts, each containing 25% of the data. The first quartile (Q1) marks the 25th percentile, the second quartile (Q2) is the median (50th percentile), and the third quartile (Q3) marks the 75th percentile. Deciles are values that divide a data set into ten equal parts, each containing 10% of the data. For example, the first decile (D1) marks the 10th percentile, and the ninth decile (D9) marks the 90th percentile. Parametric refers to a type of statistical test or model that makes specific assumptions about the parameters of the population distribution from which the data are drawn, often assuming the data follow a normal distribution. Send me the next one 📸