In this assignment, we will be exploring various concepts in statistical analysis and applying them to different scenarios. There are three parts to this assignment, each with its own set of questions and problems. In Part A, we will be working with descriptive statistics and probability. In Part B, we will be calculating various measures of central tendency and variability. Finally, in Part C, we will be exploring hypothesis testing and inferential statistics.

Part A:

The first set of questions in Part A focuses on descriptive statistics. Descriptive statistics are used to summarize and describe the main features of a dataset. They help us understand the distribution of data and provide insights into the characteristics of the population we are studying.

Question 1: Suppose we have a dataset with the following values: 2, 4, 6, 8, and 10. Calculate the mean, median, and mode for this dataset.

To calculate the mean, we sum up all the values and divide by the total number of values. In this case, the mean would be (2+4+6+8+10)/5 = 6.

The median is the middle value of a dataset when it is ordered from smallest to largest. In this case, the median would be 6.

The mode is the value that appears most frequently in the dataset. In this case, there are no repeating values, so there is no mode.

Question 2: Suppose we have another dataset with the following values: 1, 2, 3, 4, 5, and 6. Calculate the range, variance, and standard deviation for this dataset.

The range is the difference between the largest and smallest values in a dataset. In this case, the range would be 6-1 = 5.

To calculate the variance, we first need to find the mean of the dataset, which would be (1+2+3+4+5+6)/6 = 3.5. Then, we subtract the mean from each value, square the result, sum up all the squared values, and divide by the total number of values. The variance for this dataset would be [(1-3.5)^2 + (2-3.5)^2 + (3-3.5)^2 + (4-3.5)^2 + (5-3.5)^2 + (6-3.5)^2]/6 = 2.92.

The standard deviation is the square root of the variance. In this case, the standard deviation would be the square root of 2.92, which is approximately 1.71.

Question 3: Probability is used to quantify the likelihood of a particular event occurring. Suppose we have a biased coin that lands on heads 80% of the time and tails 20% of the time. What is the probability of getting three heads in a row when flipping this coin?

To calculate this probability, we need to multiply the probabilities of each individual event. In this case, the probability of getting heads three times in a row would be (0.8)*(0.8)*(0.8) = 0.512, or 51.2%.

Part B:

The second set of questions in this assignment focuses on measures of central tendency and variability. These measures provide us with insights into the typical values and spread of a dataset.

Question 1: Suppose we have a dataset with the following values: 10, 15, 20, 25, and 30. Calculate the mean, median, and mode for this dataset.

As before, the mean can be calculated by summing up all the values and dividing by the total number of values. In this case, the mean would be (10+15+20+25+30)/5 = 20.

The median is the middle value, so in this case, the median would be 20.

Since there are no repeating values, there is no mode for this dataset.

Question 2: Suppose we have another dataset with the following values: 5, 10, 15, 20, 25, and 30. Calculate the range, variance, and standard deviation for this dataset.

The range can be calculated by finding the difference between the largest and smallest values. In this case, the range would be 30-5 = 25.

To find the variance, we first need to calculate the mean, which in this case would be (5+10+15+20+25+30)/6 = 17.5. Then, we subtract the mean from each value, square the result, sum up all the squared values, and divide by the total number of values. The variance for this dataset would be [(5-17.5)^2 + (10-17.5)^2 + (15-17.5)^2 + (20-17.5)^2 + (25-17.5)^2 + (30-17.5)^2]/6 = 62.5.

The standard deviation is the square root of the variance. In this case, the standard deviation would be the square root of 62.5, which is approximately 7.91.

Question 3: Now let’s consider a dataset with the following values: 10, 15, 20, 25, 30, and 35. Calculate the z-score for the value 25.

The z-score is a measure of how many standard deviations a particular value is away from the mean. It can be calculated by subtracting the mean from the value of interest and dividing by the standard deviation. In this case, the mean is 20 and the standard deviation is approximately 7.91. So, the z-score for the value 25 would be (25-20)/7.91 = 0.63.

Part C:

The third part of this assignment focuses on hypothesis testing and inferential statistics. In hypothesis testing, we make assumptions about a population based on a sample and use statistical tests to determine the likelihood of our assumptions being true. Inferential statistics allow us to draw conclusions about a population based on sample data.

Question 1: Suppose we have two independent samples, sample A and sample B, and we want to test if there is a significant difference between the means of the two samples. The null hypothesis states that the means of the two samples are equal, and the alternative hypothesis states that the means are not equal. We have a sample size of 20 for both samples, and the mean of sample A is 5.6, while the mean of sample B is 7.2. The standard deviation for both samples is 2.5. Use a two-sample t-test to determine if there is a significant difference between the means of the two samples, assuming equal variances.

To perform a two-sample t-test, we first need to calculate the t-value. The formula for the t-value is (mean_A – mean_B)/sqrt[(s^2_A/n_A)+(s^2_B/n_B)], where mean_A and mean_B are the sample means, s^2_A and s^2_B are the sample variances, and n_A and n_B are the sample sizes. In this case, the t-value would be (5.6-7.2)/sqrt[(2.5^2/20)+(2.5^2/20)] = -3.2/sqrt[(0.625)+(0.625)] = -3.2/0.707 = -4.53.

We then compare the t-value to the critical value at a specific significance level α. Assuming a two-tailed test with α = 0.05, the critical value would be ±2.093. Since the absolute value of the t-value (-4.53) is greater than the critical value (2.093), we can reject the null hypothesis and conclude that there is a significant difference between the means of the two samples.

Question 2: Now let’s consider a one-sample t-test. Suppose we have a sample of 50 students, and we want to test if the average score of the student population is significantly different from 70. The mean score of the sample is 72, and the standard deviation is 5. Use a one-sample t-test to determine if there is a significant difference between the mean score of the student population and 70.

To perform a one-sample t-test, we first calculate the t-value, which is given by (mean_sample – μ)/(s/sqrt(n)), where mean_sample is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. In this case, the t-value would be (72-70)/(5/sqrt(50)) = 2/(5/sqrt(50)) = 2/(5/7.07) = 2/0.707 = 2.83.

We then compare the t-value to the critical value at a specific significance level α. Assuming a two-tailed test with α = 0.05, the critical value would be ±2.009. Since the absolute value of the t-value (2.83) is greater than the critical value (2.009), we can reject the null hypothesis and conclude that the mean score of the student population is significantly different from 70.

In conclusion, this assignment has allowed us to practice and apply various concepts in statistical analysis, including descriptive statistics, measures of central tendency and variability, hypothesis testing, and inferential statistics. By working through the different questions and problems, we have gained a deeper understanding of how these concepts are used to analyze and interpret data.