Probability and Statistics
A random sample of 27 observations from a large population has a mean of 22 and a standard deviation of 4.8. Can we conclude at that the population mean is significantly below 24?
Suppose you run a website with an off-white background and the mean time spent on it is 20 minutes. You consider changing the background to yellow to increase time spent. Set up the null hypothesis that the change has no effect on the mean time, and an alternative hypothesis that the mean time is greater than 20 minutes due to the change. Using a sample size of 100, calculate the p-value when the sample mean time spent is 25 minutes. Determine if the null hypothesis can be rejected at a significance level of 0.05.
Assume John claims he has a technique that lets him roll a 6 more frequently than average using a special dice rolling technique. He rolls the dice a thousand times and gets a 6 in 20% of the cases. Given these results, determine if the findings are statistically significant by calculating the p-value.
A study was conducted to determine the effectiveness of a weight loss program. The table below shows the before and after weights of 10 subjects in the program. Is this program effective for reducing weight? Use a 5% significance level.
Calculate the t-value for a paired t-test given the before values, after values, and differences. The mean of the differences is 3, the standard deviation is 4.69, and there is a sample size of 8.
How many ways can you arrange 5 books on a shelf where the order is important?
How many ways can you pick 2 co-captains on a team of 20 where the order does not matter?
In a 50-person race, how many ways can you award gold, silver, and bronze where the order is important?
Determine how many permutations and combinations exist for the letters A, B, and C, where order matters for permutations and does not matter for combinations.
Given and , calculate the number of permutations and combinations using the formulas for permutations and combinations.
Discuss the concept of a probability mass function (PMF) for a discrete variable using a six-sided dice example, where each outcome has a probability of rac{1}{6}.
Explain the cumulative distribution function (CDF) for the given discrete variable example and how it relates to cumulative probability.
Describe the concept of a rigged six-sided dice where threes and fours cannot be rolled, affecting the probability mass function (PMF) and cumulative distribution function (CDF).
Illustrate a continuous distribution example using women's heights, explaining the probability density function (PDF) and cumulative distribution function (CDF).
Link the PDF and CDF by discussing how the gradient of the CDF corresponds to the PDF for continuous distributions.
What is the probability that the time between two customer arrivals is less than five minutes?
We have a sample of 40 packages of rice with a mean weight of 5.7 oz and a standard deviation of 0.4 oz. Find the best estimate of the population mean.
Find the best point estimate for the mean of the population given the mean times for the 100-yard dash.
Suppose that we have a sample of data with values 5, 8, 10, 7, 10, and 14. Find the point estimate of the population mean.
Calculate the point estimate of the population standard deviation given a sample with values 5, 8, 10, 7, 10, and 14.