Probability and Statistics

What is the main weight of all the apples in the orchard?

The scatter plot shows the relationship between the number of slices of pizza eaten by each member of a football team and the number of laps around the block the player could run immediately after. The equation of the regression line is shown in the graph: $\hat{y} = 10 - 0.67x$. Interpret the slope and y-intercept.

A scatter plot shows the relationship between the amount of sugar added to water and the freshness of flowers. If the regression line is given by the equation $\hat{y} = 180.8 + 15.8x$, interpret the slope and y-intercept.

Interpret the slope and intercept of the least squared regression line given by the equation $\hat{\text{score}} = 52.3 + 10.5 \times \text{hours}$, where 'hours' represents the number of hours students studied for a test.

Find the marginal probability density function (PDF) of X given the joint probability density function of two continuous random variables, X and Y, is $f(x, y) = \frac{2}{3}x + 2y$ for $x \leq 1$ and $y$ between $Z$ and 1, and zero elsewhere.

Find the probability that $X \leq 0.5$ and $Y \leq 0.5$ given the joint probability density function of two continuous random variables, X and Y, is $f(x, y) = \frac{2}{3}x + 2y$ for $x \leq 1$ and $y$ between Z and 1, and zero elsewhere.

Given two random variables X and Y with a joint distribution as listed in the provided table, find the marginal distribution of X and Y, and calculate the expected values of X and Y.

Using the least squares method, find the equation of the line that best fits the given set of data points (x, y): (1, 1.5), (2, 3.8), (3, 6.7), (4, 9.0), (5, 11.2), (6, 13.6), (7, 16). Calculate the slope and the y-intercept of the line of best fit.

Using Excel, calculate the slope and y-intercept of the line of best fit for the given data points.

Calculate the least squares regression equation $\hat{y} = \beta_0 + \beta_1 x$ for the given small data set by hand. Find the estimates for the intercept $\beta_0$ and the slope $\beta_1$.

Given a set of data where the X values represent the number of questions correct out of a possible 20, and the Y values represent students' attitude percentages towards taking tests, use Simple Linear Regression to predict a student's attitude (Y) given a score (X). Specifically, calculate the slope (b) of the regression line using Pearson's correlation coefficient and standard deviations, determine the Y intercept (a), and use these to form the regression equation $y = a + bx$.

Given a training data set containing values for independent variable $x$ (e.g., height of a person) and dependent variable $y$ (e.g., weight), use linear regression to find the linear function $g(x) = \alpha x + \beta$ that best predicts $y$ from $x$.

This involves finding the values of $\alpha$ (slope) and $\beta$ (intercept) that minimize the sum of squared differences between the observed values and the values predicted by the function.

An online poll asked a sample of 850 adult Americans whether they watched any World Cup soccer games. 62% of respondents said they did. What is the Margin of Error for this poll, rounded to the nearest tenth of a percent? What is the 95% confidence interval for the percent of all adult Americans who watched the World Cup?

What is the probability that our sample proportion of 0.54 is within two standard deviations of the population proportion?

With 95% confidence, what is the confidence interval for the proportion of voters supporting candidate A?

Find the marginal distribution for the gender variable. Calculate the total boys and total girls, and express them as percentages of the total sample size.

Find the marginal distribution for the color preference. Calculate the total for each color (red, blue, green), and express them as percentages of the total sample size.

In a classroom of 200 students, analyze the relationship between the amount of time studied and the percentage of correct answers. Given a two-way table, focus on calculating and interpreting marginal and conditional distributions.

Simplified binomial random variable problem: We have a binomial random variable with parameters $N$ and $\theta$. You flip a coin $n$ times with $\theta$ as the probability of heads at each toss. After flipping, you observe a numerical value $k$ for random variable $K$. Estimate $\theta$ using maximum likelihood methodology.

Independent identically distributed normal variables problem: Given $n$ independent identically distributed normal random variables with unknown mean mu and variance $v$, estimate mu and $v$ from these observations using maximum likelihood estimation.