Based on the sample data, what is the probability that a batch exceeds the sample mean? Use the T-distribution table to calculate this. Note that in the midterm and final, you are only allowed calculators and the physical, printed tables.  

The expectation of the difference of two independent random variables is equal to the difference of their expectations, but the variance of the difference is equal to the sum of their variances. 

Should the machine be replaced if the factory’s rule is to replace it when 20% or more of batches exceed the sample mean?  

A factory monitors the time (in minutes) it takes for machines to complete a batch of products in a manufacturing process. From past data and supplier specifications, it is known that the average completion time of this machine is 50 minutes. However, employees have noticed that in recent weeks, the machine has been taking more time to complete one batch. So, they decided to take a random sample of 16 batches of products. They measured the time it takes for the machine to complete each batch. After collecting and summarizing the data, they found the following about the 16 batches:

Suppose you calculate a 95% confidence interval for the mean weight of apples in a batch as (150, 170) grams. You then repeat the experiment with a larger sample size but with the same sample mean and standard deviation. Which of the following statements is correct about the new confidence interval?

Let X∼N(5,4)X \sim N(5, 4) and Y∼N(3,9)Y \sim N(3, 9) where XX and YY are independent random variables. Calculate the expectation and variance of the linear combination Z=3X−2YZ = 3X – 2Y.  Note: X∼N(5,4)X \sim N(5, 4) – this notation is saying that X is distributed as normal with mean 5 and variance 4. Same notation applies to Y.

The Student T distribution has heavier tails than the standard normal distribution, meaning it places more probability in the tails.

A researcher wants to calculate a two-sided 95% confidence interval for the mean time it takes to complete a task. In one study, they take a small sample (n=12) with a sample mean of 30 minutes and a sample standard deviation of 5. In another study, they take a large sample (n=100) with the same sample mean and standard deviation. Which of the following correctly describes the difference between the two confidence intervals?

The sum of the probabilities of all possible outcomes in a sample space is always 1.  

A researcher calculates the margin of error for a 95% confidence interval for the mean height of a plant species. The sample mean is 20 cm, the sample standard deviation is 4 cm, and the sample size is 16. Using the T-distribution, what is the margin of error of the confidence interval? Choose the best/closest answer.