The quality-control manager of a candy company wants to discover whether a filling machine over-or underfills 16-ounce bags randomly. The following data represent the filling stat us of 18 consecutive bags: A A A A A R R R A A A A A A A A A A. A bag is rejected (R) if it is either overfilled or underfilled and accepted (A) if it is filled according to specification. Test the randomnes of the filling machine in the way that it over- or underfills at the $\alpha$=0.05 level of significance.

Use Newton’s Method to approximate, to three decimal places, the x-coordinate of the point of intersection of the graphs of the two equations. y = ln x, y = 3 - x

A die is rolled repeatedly until a 6 falls uppermost. Let the random variable X denote the number of times the die is rolled. What are the values that X may assume?

Web crawlers need to estimate the frequency of changes to Web sites to maintain a current index for Web searches. Assume that the changes to a Web site follow a Poisson process with a mean of 3.5 days. a. What is the probability that the next change occurs in less than 2.0 days? b. What is the probability that the time until the next change is greater 7.0 days? c. What is the time of the next change that is exceeded with probability 90%? d. What is the probability that the next change occurs in less than 10.0 days, given that it has not yet occurred after 3.0 days?

The distribution of the time until a Web site changes is important to Web crawlers that search engines use to maintain current information about Web sites. The distribution of the time until change (in days) of a Web site is approximated in the following table.

$$\begin{array}{cc}\text { Days until Changes } & \text { Probability } \\ 1.5 & 0.05 \\ 3.0 & 0.25 \\ 4.5 & 0.35 \\ 5.0 & 0.20 \\ 7.0 & 0.15\end{array}$$

Calculate the probability mass function of the days until change.

An article in Knee Surgery Sports Traumatology, Arthroscopy showed a mean time of 129 minutes and a standard deviation of 14 minutes for ACL reconstruction surgery for high-volume hospitals (with more than 300 such surgeries per year). If a high-volume hospital needs to schedule 10 surgeries, what are the mean and variance of the total time to complete these surgeries? Assume that the times of the surgeries are independent and normally distributed.

What's the probability that at least one will have twins if all were taking Clomid?

What's the probability that at least 3 will be on Snapchat?

What's the probability that at least one will have twins if half were taking Clomid?

Suppose that a coin is flipped and a die is rolled. Let $E_1$ denote the event "the coin shows a tail," let $E_2$ denote the event "the die shows a 3," and let $E_3$ denote the event "the coin shows heads and the die shows an odd number." List the elements of the event $E_1\: or \: E_2$.

Samples of skin experiencing desquamation are analyzed for both moisture and melanin content. The results from 100 skin samples are as follows:

Let $A$ denote the event that a sample has low melanin content, and let $B$ denote the event that a sample has high moisture content. Determine the following probabilities:

$$P(A)$$

Prove that the length of the curve r=$= f ( \theta )$, for $\alpha \leq \theta \leq \beta$, is $L = \int _ { \alpha } ^ { \beta } \sqrt { f ( \theta ) ^ { 2 } + f ^ { \prime } ( \theta ) ^ { 2 } } d \theta$.

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:

$$\begin{array}{llcc} & & \text { Shock Resistance }\\ & & \text { High } & \text { Low } \\ \text { Scratch } & \text { High } & 70 & 9 \\ \text { Resistance } & \text { Low } & 16 & 5 \end{array}$$

Let $A$ denote the event that a disk has high shock resistance, and let $B$ denote the event that a disk has high scratch resistance. If a disk is selected at random, determine the following probabilities:

$$P(B)$$

Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows:

$$\begin{array}{llcc} & & \text { Shock Resistance }\\ & & \text { High } & \text { Low } \\ \text { Scratch } & \text { High } & 70 & 9 \\ \text { Resistance } & \text { Low } & 16 & 5 \end{array}$$

Let $A$ denote the event that a disk has high shock resistance, and let $B$ denote the event that a disk has high scratch resistance. If a disk is selected at random, determine the following probabilities:

$$P\left(A^{\prime}\right)$$