For any cumulative distribution function F(x), show that if $a \leq b,$ then $F(a) \leq F(b)$

Use integration by parts to find each integral. $\int s(2 s+1)^{4} d s$

Do seagulls show a preference for where they land? To answer this question, biologists conducted a study in an enclosed outdoor space with a piece of shore made up of 56% sand, 29% mud, and 15% rocks. The biologists chose 200 seagulls at random. Each seagull was released into the outdoor space on its own and observed until it landed somewhere on the piece of shore. In all, 128 seagulls landed on the sand, 61 landed in the mud, and 11 landed on the rocks. Calculate the value of the chi-square test statistic for these data.

A sales clerk’s weekly salary is $300 plus 2% of her total week’s sales. Find a function P(x) for her pay for a week in which she sold x dollars of merchandise.

What is the probability that the student has allergies? $\text {(a) } \frac{19}{88} \text { (b) } \frac{34}{88} \text { (c) } \frac{54}{88} \text { (d) } \frac{34}{54}$

The number of cars arriving at a given intersection follows a Poisson distribution with a mean rate of 4 per second. a. What is the probability that 3 cars arrive in a given second? b. What is the probability that 8 cars arrive in three seconds? c. What is the probability that more than 3 cars arrive in a period of two seconds?

Does the infinite geometric series 1,000,000 + 100,000 + 10,000 + ... converge or diverge?

Differentiate each function. $f(z)=e^{z+\cos z}$

Calculate the relevant critical value for each of the following settings, assuming the conditions are met: You want to calculate a 90% confidence interval for a population mean using a sample of size 95.

Briefly explain the citric acid cycle.

Draw a tree diagram to display all the possible head–tail sequences that can occur when you flip a coin three times.

When the sample size is small relative to the population size, there is little difference between sampling with and without replacement. Explain in your own words why that statement is true.

A new muscle relaxant is available. Researchers from the firm developing the relaxant have done studies that indicate that the time lapse between administration of the drug and beginning effects of the drug is normally distributed, with mean $\mu=38$ minutes and standard deviation $\sigma=5$ minutes. The drug is administered to one patient selected at random. What is the probability that the time it takes to go into effect is 35 minutes or less?

Six measurements were made of the mineral content (in percent) of spinach, with the following results.

$$\begin{matrix} \text{19.1} & \text{20.8} & \text{20.8} & \text{21.4} & \text{20.5} & \text{19.7}\\ \end{matrix}$$

Assuming the population to be normally distributed, construct a 99% confidence interval for population variance $\sigma^{2}.$