A researcher suspected that the number of betweenmeal snacks...

A researcher suspected that the number of betweenmeal snacks eaten by students in a day during final examinations might depend on the number of tests a student had to take on that day. The accompanying table shows joint probabilities, estimated from a survey. $$ \begin{array}{ccccc} \hline \text { Number of } & \underline {\text { Number of Tests }(\mathrm{X})} \\ \text { Snacks }(Y) & 0 & 1 & 2 & 3 \\ \hline 0 & 0.07 & 0.09 & 0.06 & 0.01 \\ 1 & 0.07 & 0.06 & 0.07 & 0.01 \\ 2 & 0.06 & 0.07 & 0.14 & 0.03 \\ 3 & 0.02 & 0.04 & 0.16 & 0.04 \\ \hline \end{array} $$ a. Find the probability distribution of $X$ and compute the mean number of tests taken by students on that day. b. Find the probability distribution of $Y$ and, hence, the mean number of snacks eaten by students on that day. c. Find and interpret the conditional probability distribution of $Y$, given that $X=3$. d. Find the covariance between $X$ and $Y$. e. Are number of snacks and number of tests independent of each other?

A researcher suspected that the number of betweenmeal snacks eaten by students in a day during final examinations might depend on the number of tests a student had to take on that day. The accompanying table shows joint probabilities, estimated from a survey.
$$
\begin{array}{ccccc}
\hline \text { Number of } & \underline {\text { Number of Tests }(\mathrm{X})} \\
\text { Snacks }(Y) & 0 & 1 & 2 & 3 \\
\hline 0 & 0.07 & 0.09 & 0.06 & 0.01 \\
1 & 0.07 & 0.06 & 0.07 & 0.01 \\
2 & 0.06 & 0.07 & 0.14 & 0.03 \\
3 & 0.02 & 0.04 & 0.16 & 0.04 \\
\hline
\end{array}
$$
a. Find the probability distribution of $X$ and compute the mean number of tests taken by students on that day.
b. Find the probability distribution of $Y$ and, hence, the mean number of snacks eaten by students on that day.
c. Find and interpret the conditional probability distribution of $Y$, given that $X=3$.
d. Find the covariance between $X$ and $Y$.
e. Are number of snacks and number of tests independent of each other?

Key Concepts

Statistical Independence

Two random variables are statistically independent if the occurrence of one does not affect the probability distribution of the other. In terms of probability, this means that the joint probability distribution is the product of the marginal distributions. Testing for independence helps determine whether the variables have any association beyond random chance.

Conditional Probability Distribution

A conditional probability distribution gives the probabilities of outcomes for one random variable given that another random variable is fixed at a particular value. This concept helps in understanding how the probability distribution of one variable changes when additional information about another variable is available.

Covariance

Covariance is a measure that indicates the extent to which two random variables change together. A positive covariance suggests that the variables tend to increase together, while a negative covariance indicates that as one increases, the other tends to decrease. It provides insights into the linear relationship between the variables.

Marginal Probability Distribution

The marginal probability distribution of a variable is obtained by summing or integrating the joint probability distribution over all the other variables. This distribution reflects the probabilities of the individual outcomes of that variable without consideration of the others.

Joint Probability Distribution

A joint probability distribution describes the probability that two or more random variables take on specific values at the same time. It is represented in a table or function that assigns a probability to each combination of outcomes, thereby providing a complete picture of the relationship between the variables involved.

Expected Value

The expected value, or mean, of a random variable is a measure of its central tendency. It is computed as the weighted sum of all possible values of the variable, where the weights are their corresponding probabilities. The expected value provides a summary measure of the typical value in the distribution.

Transcript

00:01 Hello, so here we let x denote the time spent studying by students in the week before final exams.

00:06 So the random variable x here follows a normal distribution with sigma is equal to 8, and the number of students n is equal to 4.

00:13 So the standard error of the sampling distribution of the sample mean, that's sigma sub x bar, which is going to be equal to sigma divided by the square root of n.

00:23 So that's 8 divided by the square root of 4 or 8 over 4 or 8 over 2, which is going to be equal to four.

00:31 So we then have for part a here that the sample mean exceeds the population mean by more than two hours.

00:40 So we have the probability that x bar is greater than mu plus two, which is going to be equal to the probability of x bar minus mu, which is greater than 2.

01:00 Gives us that the probability that x bar minus mu over sigma is greater than 2 over 4.

01:07 So we have the probability then that z is going to be greater than 0 .5, giving us the probability this is equal to 1 minus the probability that z is less than or equal to 0 .5, which is going to be equal to 1 minus 0 .6915, which is going to be equal to 0 .30, which is going to be equal to 0 .30.

01:30 So therefore, the probability that the sample mean exceeds the population mean by more than two hours is going to be 0 .3085.

01:44 Then for part b, we want the sample mean more than three hours below the population means.

01:52 So we want the probability that mu minus x bar is going to be greater than three...

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