The amount of contaminants that are allowed in food products is determined by the FDA (Food and Drug Administration). Common contaminants in cow milk include feces, blood, hormones, and antibiotics. Suppose you work for the FDA and are told that the current amount of somatic cells (common name "pus") in 1 cc of cow milk is currently 750,000 (note: this is the actual allowed amount in the US!). You are also told the standard deviation is 123000 cells. The FDA then tasks you with checking to see if this is accurate.

You collect a random sample of 55 specimens (1 cc each) which results in a sample mean of 782227 pus cells. Use this sample data to create a sampling distribution. Assume that the population mean is equal to the FDA's legal limit and see what the probability is for getting your random sample.

a. Why is the sampling distribution approximately normal?

b. What is the mean of the sampling distribution?

c. What is the standard deviation of the sampling distribution?

d. Assuming that the population mean is 750,000, what is the probability that a simple random sample of 55 1 cc specimens has a mean of at least 782227 pus cells?

e. Is this unusual? Use the rule of thumb that events with probability less than 5% are considered unusual.

f. Explain your results above and use them to make an argument that the assumed population mean is incorrect. (6 points) Structure your essay as follows:
Describe the population and parameter for this situation.

Describe the sample and statistic for this situation.

Give a brief explanation of what a sampling distribution is.

Describe the sampling distribution for this situation.

Explain why the Central Limit Theorem applies in this situation.

Interpret the answer to part d.

Use the answer to part e. to argue that the assumed population mean is either correct or incorrect. If incorrect, indicate whether you think the actual population mean is greater or less than the assumed value.

Explain what the FDA should do with this information.

A team of visiting polio eradication workers were informed during their orientation session that population-wide studies done in their host country showed that the risk of polio in villages of that country was strongly epidemiologically associated with the village’s economic/human development circumstances, which ranged greatly from village to village.  In some villages, residents lived in hand-constructed huts with no running water, no latrines or sewage disposal areas, and no electricity.  In other places, residents lived in wooden or adobe homes which, though modest by Western standards, had all of the above services in place and whose street side craft shops and food markets did a brisk business, catering both to locals and visitors.

Knowing this information, the team went into several villages and attempted to assign a “human development rating” to each family.  This was based on that family’s income situation, access to running water, access to elementary school for their children, and the condition of the home. To their surprise, they found that families in all the villages had no difference in polio risk based on the family’s human development rating.

Are steers and heifers equally distributed between angus and herefords

In a test of the Atkins weight loss program, 41 individuals participated in a randomized trial with overweight adults. After 12 months, the mean weight loss was found to be 2.1 Kg, with a standard deviation of 4.8 Kg. (a) What is the best point estimate for the mean weight loss of all overweight adults who follow the Atkins program? (b) Construct a 90% confidence interval for the mean weight loss for all such subjects? (c) Test the hypothesis that the mean weight loss is 3.2 Kg. (d) In part (c), could we conclude the test simply by looking at the confidence interval constructed in part (b)? Explain. Also, what is the minimum mean weight loss that would be rejected by the sample data? (e) Suppose that a weight loss program is considered effective only if the weight loss is at least 3.2 Kg after 12 months. Do you think, the Atkinson program seems to effective at 90% confidence level? (f) For part (e), could we use the confidence interval constructed in part (b)?

1. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 234 feet and a standard deviation of 58 feet. We randomly sample 49 fly balls.

b) What is the probability that the 49 balls traveled an average of less than 226 feet? (Round your answer to four decimal places.)

c) Find the 60th percentile of the distribution of the average of 49 fly balls. (Round your answer to two decimal places.)

4.The length of songs in a collector's iTunes album collection is uniformly distributed from two to 3.7 minutes. Suppose we randomly pick five albums from the collection. There are a total of 44 songs on the five albums.
d) Give the distribution of X (Round your answers to four decimal places.)

X ~ N ( 2.85 , ? )

e) Find the first quartile for the average song length. (Round your answer to two decimal places.)

f) Find the IQR (interquartile range) for the average song length. (Round your answer to two decimal places.)

6.The percent of fat calories that a person consumes each day is normally distributed with a mean of about 34 and a standard deviation of about ten. Suppose that 25 individuals are randomly chosen. Let

X = average percent of fat calories.

(a) Give the distribution of X. (Round your standard deviation to two decimal places.)

X ~ N ( 34, ? )

(c) Find the first quartile for the average percent of fat calories. (Round your answer to two decimal places.)

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,300. We randomly survey ten teachers from that district. (Round your answers to the nearest dollar.)

(a) Find the 90^th percentile for an individual teacher's salary.

(b) Find the 90^th percentile for the average teacher's salary.

The function f ( x ) = 2 x 3 − 39 x 2 + 180 x + 3 f ( x ) = 2 x 3 - 39 x 2 + 180 x + 3 has one local minimum and one local maximum. Use a graph of the function to estimate these local extrema. This function has a local minimum at x x = with output value: and a local maximum at x x = with output value. my open math.

University of Minnesota recently developed a new apple called First Kiss. At the Minnesota State Fair, 200 random fair goers sampled both the new apple and a Honeycrisp apple and asked which apple they preferred. Of the 200 people, 134 people preferred First Kiss and the remaining 66 people preferred Honeycrisp. Is there any preference between the two apples? Use α =0.05 to perform a 5-step test of hypothesis.

Suppose you pick 50 clovers from the grass in Washington square park. Some clovers are four-leaf clovers but most have only three leaves. Suppose clovers develop their leaves inde- pendently of each other and there is some probability p that they will develop into a four-leaf clover.

Calculate the probability that you get two or fewer four-leaf clovers, for each of p = 0.1, 0.05, 0.01. For each value of p, do the calculation in three ways: the exact calculation, the Poisson ap- proximation, and the Normal approximation (with the histogram correction.) Comment on your results – which approximation is better when?

A bottled water distributor wants to estimate the amount of water contained in 1​-gallon bottles purchased from a nationally known water bottling company. The water bottling​ company's specifications state that the standard deviation of the amount of water is equal to 0.03 gallon. A random sample of 50 bottles is​ selected, and the sample mean amount of water per 1​-gallon bottle is 0.941 gallon. Complete parts​ (a) through​ (d).

a. Construct a 99​% confidence interval estimate for the population mean amount of water included in a​ 1-gallon bottle. nothingless than or equalsmuless than or equals nothing

b. On the basis of these​ results, do you think that the distributor has a right to complain to the water bottling​ company? Why?

c. Must you assume that the population amount of water per bottle is normally distributed​ here? Explain.

d. Construct a 90​% confidence interval estimate. How does this change your answer to part​ (b)?

Before 1918, approximately 60% of the wolves in a region were male, and 40% were female. However, cattle ranchers in this area have made a determined effort to exterminate wolves. From 1918 to the present, approximately 65% of wolves in the region are male, and 35% are female. Biologists suspect that male wolves are more likely than females to return to an area where the population has been greatly reduced. (Round your answers to three decimal places.) (

a) Before 1918, in a random sample of 11 wolves spotted in the region, what is the probability that 8 or more were male?

What is the probability that 8 or more were female?

What is the probability that fewer than 5 were female?

b) For the period from 1918 to the present, in a random sample of 11 wolves spotted in the region, what is the probability that 8 or more were male?

What is the probability that 8 or more were female?

What is the probability that fewer than 5 were female?

The price to earnings ratio (P/E) is an important tool in financial work. A random sample of 14 large U.S. banks (J. P. Morgan, Bank of America, etc) gave the following P/E ratios.

24, 16, 22, 14, 12, 13, 17, 22, 15, 19, 23, 13, 11, 18

Generally speaking, a low P/E ratio indicates a "value" or bargain stock. Financial publications indicated that the P/E ratio of the S&P 500 stock index has typically been 20.0. Let x be a random variable representing the P/E ratio of all large U.S. bank stocks. We assume that x has a normal distribution and σ = 4.1. Do these data indicate that the P/E ratio of all U.S. bank stocks is less than 20.0? Use α = 0.05.

(a) Enter the following. x =

s = (

b) Identify the claim, the null hypothesis, and the alternative hypothesis.

Claim: 20.0

Ho: 20.0

H1: 20.0

(c) Will you use a left-tailed, right-tailed, or two-tailed test? two-tailed left-tailed right-tailed

d) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since we assume that x has a normal distribution with known σ.

The standard normal, since n is large with known σ.

The standard normal, since we assume that x has a normal distribution with known σ.

The standard normal, since we assume that x has a normal distribution with unknown σ.

The standard normal, since n is large with unknown σ. The Student's t, since n is large with unknown σ.

(e) Sketch the sampling distribution showing the area corresponding to the approximate P-value.

(f) Will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

    At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant

.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(g) State your conclusion in the context of the application.

Fail to reject the null hypothesis, there is insufficient evidence that average P/E for large banks is less than the S&P 500 Index.

Fail to reject the null hypothesis, there is sufficient evidence that average P/E for large banks is less than the S&P 500 Index.    

  Reject the null hypothesis, there is insufficient evidence that average P/E for large banks is less than the S&P 500 Index

.Reject the null hypothesis, there is sufficient evidence that average P/E for large banks is less than the S&P 500 Index.

Let X have a binomial distribution with parameters

n = 25

and p. Calculate each of the following probabilities using the normal approximation (with the continuity correction) for the cases

p = 0.5, 0.6, and 0.8

and compare to the exact binomial probabilities calculated directly from the formula for

b(x; n, p).

(Round your answers to four decimal places.)

(a)

P(15 ≤ X ≤ 20)

Write the bivariate normal pdf f(x, y; θ1, θ2, θ3, θ4, θ5) in exponential form and show that Z1 = n i=1 X2 i , Z2 = n i=1 Y2 i , Z3 = n i=1 XiYi, Z4 = n i=1 Xi, and Z5 = n i=1 Yi are joint sufficient statistics for θ1, θ2, θ3, θ4, and θ5.

Explain the relationship between types or levels of data and types of statistical analyses available to such levels of data. Feel free to illustrate with examples

1.A. Dr. Smith purchased a network attached storage server with eleven 2TB hard drives in it. When he purchased the system, unknowingly to him, the supplier used hard drives from a bad manufacturing batch and thus among his drives there are four underperforming drives. (Underperforming drives have a much shorter lifetime.) They figured this out and since there is no way of determining which hard drives are faulty ahead of time (i.e., ahead of their death), they issued Dr. Smith a full refund and told him to keep the drives. He wanted to trash the drives but one of his students requested a few of them for his project. The student figured to use the drives in a RAID-1 configuration. (From Wikipedia : “A RAID 1 creates an exact copy (or mirror) of a set of data on two or more disks. This is useful when read performance or reliability is more important than data storage capacity.”) What are the chances that he is not going to lose any data prematurely if Dr. Smith gave him two, three, and four of the drives (what’s the probability for each of those setups)?

1.B. Same situation as in 1.A. However, the student is more adventurous, he wants to use RAID-5. (RAID 5 is a redundant array of independent disks configuration that uses disk striping with parity. Because data and parity are striped evenly across all of the disks, no single disk is a bottleneck. Striping also allows users to reconstruct data in case of a disk failure. RAID 5 evenly balances reads and writes, and is currently one of the most commonly used RAID methods. It has more usable storage than RAID 1 and RAID 10 configurations, and provides performance equivalent to RAID 0. RAID 5 groups have a minimum of three hard disk drives (HDDs) and no maximum. Because the parity data is spread across all drives, RAID 5 is considered one of the most secure RAID configurations.) How big of a virtual drive will he obtain and what are the chances that he is not going to lose any data prematurely if Dr. Smith gives him three or four of drives?
1.C. We are doing real time processing of data from 40 data sources. At the beginning of each time slot, each data source may (or may not) generate a data-set to be processed; the probability that any individual source actually generates a data-set is: 0.002 (and data sources are independent). In each time slot we can process up to two data-sets. As the processing is real time the processed results are only interesting if they are done within the time slot. What is the probability that we can process all incoming data-sets in any particular time slot?