Probability and Statistics

Given a uniform distribution ranging from 0 to 1, collect 20 random samples and calculate the mean of these samples. Repeat this to collect additional samples and calculate more means, then draw a histogram of these means. Discuss how the means are distributed after multiple iterations.

Using an exponential distribution, collect 20 random samples and calculate the mean of these samples. Repeat the process to collect more samples and calculate more means, then draw a histogram of these means. Analyze the distribution of these means after several iterations.

Company XYZ manufactures laptops. For quality control, two sets of laptops were tested. In the first group, 32 out of 800 were found to contain some sort of defect. In the second group, 30 out of 500 were found to have a defect. Is the difference between the two groups significant? Use a significance level of 0.05.

Patients were randomly assigned to either duct tape treatment or traditional treatment. Out of 100 warts treated with liquid nitrogen, 60 were successfully removed, and out of 104 warts treated with duct tape, 88 were successfully removed. Test if duct tape is more effective for removing warts at $\alpha = 0.01$.

Construct a 99% confidence interval for the difference between the two proportions in the effectiveness of duct tape versus liquid nitrogen in removing warts.

If P(A) is 0.34, P(B) is 0.50, and P(A \cup B) is 0.70, what is the probability of A, given that B has occurred?

What is the probability of $A \cup B$, given that event $C$ occurs?

If one of these 523 cases is randomly selected, what is the probability the person was female?

Given the cause was cardiovascular in nature, what is the probability the person was female?

What is the probability the death was cardiovascular in nature, given the person was female?

Given the cause was cerebral or respiratory in nature, what is the probability the person was male?

What is the conditional probability of $A \cap C$, given $B \cap C$?

What is the probability of $B^c$, given $A \cup C$?

Is the probability of A, given B equal to, less than, or greater than the unconditional probability of A?

If event A is a subset of event B and P(A) > 0, what can be said of P(A|B) and P(B|A)?

Given that A is a subset of B and $P(A) > 0$, what can be said about the probability of A given B?

Given the events A (Rahul eats a bagel for breakfast) and B (Rahul eats pizza for lunch) with the probabilities $P(A) = 0.6$, $P(B) = 0.5$, and the conditional probability $P(A|B) = 0.7$, find the conditional probability $P(B|A)$, rounded to the nearest hundredth.

Calculate the confidence interval for the mean value of normally distributed data using the formula: $\bar{x} \pm z \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z$ is the z-value for the desired confidence level, $s$ is the standard deviation, and $n$ is the sample size.

Given a dataset where the mean is $\bar{x}$, the sample size is $n$, and the standard deviation is $s$, determine the 95% confidence interval for the mean.

Scores on an exam are normally distributed with a population standard deviation of 5.6. A random sample of 40 scores on the exam has a mean of 32. We want to construct confidence interval estimates for the population mean at 80%, 90%, and 98% confidence levels.