Probability and Statistics

Create a comprehensive estimate for a contracting job, considering factors like project scope, weather, permits, licenses, direct and indirect costs, and desired profit margin.

A factory has a machine that dispenses 80 ml of fluid in a bottle. An employee believes the average amount of fluid is not 80 ml. Using 40 samples, he measures the average amount dispensed by the machine to be 78 ml with a standard deviation of 2.5. Part A: State the null and alternative hypotheses.

A company manufactures car batteries with an average lifespan of two or more years. An engineer believes this value to be less. Using 10 samples, he measures the average lifespan to be 1.8 years with a standard deviation of 0.5. Part A: State the null and alternative hypotheses.

Using residual plots, determine if the normality, constant variance, and linearity assumptions of a simple linear regression model are upheld for a given dataset.

Evaluate the residual plot for the activation level in the brain's pain centres versus their score on the empathic concern scale for 16 women, and determine if the assumptions of the regression model are met.

Evaluate the residual plot for the Janka hardness versus density data for 36 Australian trees, and determine if the assumptions of the regression model are met.

Calculate the residual for a point where the explanatory variable $x$ is represented by hours and the response variable $y$ is represented by the number of people attending a lecture. If the actual number of people at 3 hours is 50 and the predicted number of people at 3 hours using the best fit line is 45.46, what is the residual?

The average test score of all students in a certain class was measured to be 84 with a standard deviation of 15. If the margin of error was calculated to be 5.367 at a 95% confidence level, how many students were in this class?

Sally wants to determine what proportion of high school students are planning to enroll in college after they graduate. Given a 95% confidence level and a margin of error of 5%, what is the maximum sample size that she should use in order to perform this statistical experiment?

Suppose we wish to estimate the proportion of adults in Ontario that are in favor of harsher penalties for drug offenses. How large of a sample size is required if we wish to estimate $P$ to within $0.03$ with $95\%$ confidence?

Suppose we wish to estimate the proportion of mice that would be killed by a certain dose of a chemotherapy drug. Suppose also that previous studies have shown that approximately $10\%$ of mice are killed by this dose. How large of a sample size is required if we wish to estimate $P$ to within $0.1$ with $99\%$ confidence?

If I flip the same coin two times, what would be the probability of getting heads twice?

If a coin is tossed 2 times, find the probability of getting at least one tail.

If we flip two coins, what is the probability of not getting two tails?

Calculate the probability of flipping a tails when flipping a coin.

Suppose it is known that the heights of all Canadians follows a normal distribution with a mean of 160 centimeters and a standard deviation of 7 centimeters. What is the probability that the average height of 10 random Canadians is less than 157 centimeters?

What is the proportion of all people that have heights greater than 170 centimeters?

What is the probability a randomly selected order has at least 760 calories?

What is the probability that the mean of 9 randomly selected orders is at least 760 calories?

90% of all plant species are classified as angiosperms. If you randomly sample 200 plants from all known plant species, what is the probability that at least 95% will be flowering plants?