1. What if the size for each sample were increased to 20? Would a sample mean of 115 or more be considered unusual? Why or why not?
2. Why is the Central Limit Theorem used?
3. Consider situations in your work or home that could be addressed through a continuous probability distribution. Describe the situation and the variables and determine whether the variables are normally distributed or not. How could you change these to a normally distributed dataset?
1. To determine if a sample mean of 115 or more would be considered unusual, you can use the concept of z-scores.
If the sample size is increased to 20, the standard deviation becomes smaller due to the square root of the sample size being in the denominator of the formula for standard deviation. Consequently, the distribution of sample means will become narrower, leading to a higher chance of getting a sample mean close to the population mean.
To assess whether a sample mean of 115 or more is unusual, you can calculate the z-score using the formula:
z = (sample mean - population mean) / (standard deviation / square root of sample size)
Assuming you have the population mean and standard deviation, you can substitute those values into the formula along with the sample mean and sample size. Then, you can compare the calculated z-score with a critical value from the standard normal distribution table. If the calculated z-score is greater than the critical value (usually 1.96 for a 95% confidence level), the sample mean would be considered unusual.
2. The Central Limit Theorem is used because it provides a useful tool to make inferences about a population based on a sample. It states that when independent random variables are added, their sum tends to follow a normal distribution, regardless of the shape of the original distribution. This theorem is particularly important when dealing with large sample sizes.
The Central Limit Theorem is utilized in statistical analysis because it enables us to make assumptions about the population distribution based on the sample distribution. By assuming that the sample mean is normally distributed, we can estimate population parameters, such as the population mean, with greater accuracy and reliability.
3. In my work or home life, a situation that could be addressed through a continuous probability distribution is employee response times to customer inquiries. Let's assume we are interested in determining the average time it takes for an employee to respond to a customer query.
The variables involved in this scenario are the employee response times (measured in minutes) and the frequency of occurrences. To determine whether the response times are normally distributed, you could collect a dataset of response times and assess its distribution using various statistical techniques, such as a histogram or a normality test like the Shapiro-Wilk test.
If the response times are not normally distributed, you could consider transforming the data to achieve normality. This can be done by applying mathematical functions like logarithmic, exponential, or square root transformations. Alternatively, if the distribution is heavily skewed, you could consider using non-parametric statistical methods.
However, it is important to note that in some cases, it may not be necessary or appropriate to transform the data. It ultimately depends on the specific context and the goals of the analysis.