1. The problem statement, all variables and given/known data

This is from a practice exam:

2. Relevant equations

t-distribution?

3. The attempt at a solution

I feel like if I start with part (b) and fine the probability that 1 carton falls in the interval, I can then apply the binomial distribution to find the prob that 5 cartons fall in that interval. But I am very lost as to where to start. How can I calculate a t-value without a mu value?

Can't you use the sample mean and sample standard deviation for your t-test?

And the question doesn't ask you, "what is the probability that five randomly selected cartons will all individually fall in this interval?" It asks, "what is the probability that the average weight of five randomly selected cartons will fall in this interval." So no, not the binomial distribution.

Funny. This same exact problem was on the actual exam tonight and I still could not get it. I thought that a "T-text" made use of the fact that the random variable [tex]T = (\bar{X} - \mu)/(S/\sqrt{n})[/tex] has a t-distribution with [itex]\nu[/itex] = n - 1, right?

Let me ask this so that I know that I am interpreting the question properly: for part (b) I am being asked to find [tex]P(40.5 < \bar{X} < 42.4)[/tex] correct?

Yeah. So you let [itex] \mu [/itex] equal your sample mean, S equal your sample standard deviation. Now you can figure out what the t-score of those intervals is, and use the t-distribution to find the probability. For the bundle of 5, you just multiply the size of the t-score interval by [itex] \sqrt{5} [/itex].

I assume b means "suppose we take another carton. What is the probability that that carton's weight is in that interval?" If the question is supposed to mean "What is the probability that the population mean lies on this interval"...well, that's not an appropriate question to ask.

OK. So let's say that I calculate my t-values and I get t1 and t2. I then need to find the probability P(t1 < T < t2). Does this seem like something you would use the "t-tables" for? Or the actual PDF of the t-distribution? I ask because the t-values are tabulated but degrees of freedom AND confidence (alpha). For example, the t-tables look like that below.