Probability and Statistics

Using the Z-statistic, calculate the Z-score for a single value $x$ given the mean $\mu$ and standard deviation $\sigma$.

Calculate a T-statistic when the population standard deviation is unknown by using the sample standard deviation.

A random sample of 15 observations has a mean of 20 and a standard deviation of 3.5. To estimate the population mean with 95% confidence, determine the margin of error and the confidence interval.

With a sample size of 85, construct a 99% confidence interval for the population mean.

Suppose that $f(x)$, the PDF function, is $\frac{x}{8}$ for $x$ between 3 and 5.

Determine the following probabilities: (a) $\Pr(X < 4)$, (b) $\Pr(4 < X < 5)$, (c) $\Pr(X < 3.5 \text{ or } X > 4.5)$.

If we have a uniform distribution from A = 2 to B = 6, what is the value of the probability density function $f(X)$?

Using the PDF of an exponential distribution $f(x) = \lambda e^{-\lambda x}$, calculate the probability that $X$ is less than a given value $x$.

Calculate the correlation coefficient between the two variables given the data (x: 1, 2, 3, 4, 5, 6; y: 2, 4, 7, 9, 12, 14).

Calculate by hand the correlation coefficient $r$ for a given set of bivariate data where the $X$ values are $1, 2, 2, 3$ and the $Y$ values are $1, 2, 3, 6$.

I have three pants and four shirts. How many different outfits can I wear?

Using the counting principle, calculate the number of different outfits if I have four shirts, four pants, and four shoes.

Calculate the total number of possible solutions for a lock with 10 different numbers and five slots using the counting principle.

Create a computer password using the 26 letters of the alphabet, where each letter can be used only once, and the password is four spaces long. How many different passwords are possible?

Form a five-digit number on a given keypad for an apartment entrance. Calculate the number of possible codes if repeats are allowed.

Calculate the number of possible five-digit codes on a keypad if no repeats are allowed.

Determine the number of possible five-digit codes with at least one repeat on the keypad.

Calculate the number of possible codes if the five-digit number has to be odd.

Imagine two random variables: one depicting the height of a population in centimeters and the other depicting the weight of that population in kilograms. What is the covariance between these two variables if you observe a positive relationship as height increases, weight tends to increase as well?

What happens to the covariance if you replace height with the number of hours per week a person exercises, given that the relationship between the exercise hours and weight is negative?

Calculate the correlation for the weights measured in kilograms and grams, noting that the correlation score remains the same despite the change in scale.