R programming

Babies born in the US have birth weights that are approximately normally distributed with mean 3.339 kg and standard deviation 0.573 kg.

Using R, determine:

a) What fraction of babies are more than 1.5 standard deviations from the mean in either direction?

b) What fraction of babies are more than 1.5 kg from the mean in either direction?

c) If you took a random sample of 100 babies, what is the probability that their mean weight Y is greater than 3.5kg?

Please include the code used to obtain the answer

uppose that the longevity of a light bulb is exponential with a mean lifetime of eight years. (a) Find the probability that a light bulb lasts less than one year. (Round your answer to four decimal places.) (b) Find the probability that a light bulb lasts between six and ten years. (Round your answer to four decimal places.) (c) Seventy percent of all light bulbs last at least how long? (Round your answer to two decimal places.) yr (d) A company decides to offer a warranty to give refunds to light bulbs whose lifetime is among the lowest three percent of all bulbs. To the nearest month, what should be the cutoff lifetime for the warranty to take place? (Round your answer up to the next month.) months (e) If a light bulb has lasted seven years, what is the probability that it fails within the eighth year. (Round your answer to four decimal places.)

Suppose we know that the average height of Americans is 176cm with standard deviation of 6cm; and that the average height of Australians is 181cm with standard deviation of 8cm. Suppose in a room there are two individuals, who happen to be from the same country. We cannot hear their accents (so cannot guess in that way whether they are Americans or Australians), but we know these individuals’ heights, which are 178cm and 180cm. Assume that heights are normally distributed. Obtain the likelihood functions, and suggest the likely country of origin of these individuals.

When evaluating a chi-square test, describe the importance of the goodness of fit test. Provide an example and explain how the test is used to evaluate the data.

Arsenic occurs naturally in very low concentrations. In healthy human adults arsenic blood concentrations are approximately Normally distributed with mean 3.9 μg/dL (micrograms per decilitre) and standard deviation 1.4 μg/dL. For the purposes of this question, assume that the distribution of arsenic blood concentrations is exactly as just described.

(a) What proportion of healthy adults have arsenic blood concentrations between 2 and 4.5 μg/dL?

b) Choosing a healthy adult at random, what is the chance that their arsenic blood concentration exceeds 6.8 μg/dL?

(c) What are the lower and upper limits of the middle 80% of arsenic blood concentrations in healthy human adults?

The following table presents the average price in dollars for a dozen eggs and a gallon of milk for each month in a recent year.

Dozen Eggs	Gallon of Milk
1.94	3.58
1.80	3.52
1.77	3.50
1.83	3.47
1.69	3.43
1.67	3.40
1,65	3.43
1.88	3.47
1.89	3.47
1.96	3.52
1.96	3.54
2.01	3.58

If a linear regression model were fit, what is the value of the slope and the value of the y-intercept? Please round to 3 decimal places as necessary. Treat the price of a gallon of milk as the response variable.

3.*       Public transportation and the automobile are two methods an employee can use to get to work each day. Samples of times recorded for each method are shown. Times are in minutes.

            Public Transportation:        18   15   13   20   22   17   11   14    3   16

            Automobile:                        12   24   19   21   19   25   23    9    17 10

            a)         Compute the sample mean time to get to work for each method.

            b)         Compute the sample standard deviation for each method.

              c)         On the basis of your results from a) and b), which method of transportation should be preferred? Please explain.

For a new study conducted by a fitness magazine,

240

females were randomly selected. For each, the mean daily calorie consumption was calculated for a September-February period. A second sample of

210

females was chosen independently of the first. For each of them, the mean daily calorie consumption was calculated for a March-August period. During the September-February period, participants consumed a mean of

2385.5

calories daily with a standard deviation of

222

. During the March-August period, participants consumed a mean of

2414.5

calories daily with a standard deviation of

252.5

. The population standard deviations of daily calories consumed for females in the two periods can be estimated using the sample standard deviations, as the samples that were used to compute them were quite large. Construct a

90%

confidence interval for

−μ1μ2

, the difference between the mean daily calorie consumption

μ1

of females in September-February and the mean daily calorie consumption

μ2

of females in March-August. Then complete the table below

What is the lower limit of the 90% confidence interval?
What is the upper limit of the 90% confidence interval?

. Assume that the lifetimes (measured from the beginning of use) of lightbulbs are i.i.d. random variables with distribution P(T ≥ k) = (k + 1)−β , k = 0, 1, 2, . . . , for some β > 0. (Note that time is measured in discrete units.) In a lightbulb socket in a factory, a bulb is used until it fails, and then it is replaced at the next time unit. Let (Xn)n≥0 be the irreducible Markov chain which records the age of the bulb currently in use in the socket (Xn = 0 at times when a bulb is replaced, corresponding to a new bulb). (a) Derive the transition probabilities of the chain. (b) For each value of β, determine if the chain is positive recurrent, null recurrent, or transient.

Market research has indicated that customers are likely to bypass Roma tomatoes that weigh less than 70 grams. A produce company produces Roma tomatoes that average 78.0 grams with a standard deviation of 5.2 grams.

Suppose there were 3 undersized tomatoes in the random sample of 20. What is the probability of getting at least 3 undersized tomatoes in a random sample of 20 if the company's claim is true? Do you believe the company's claim? Why or why not?

The Chartered Financial Analyst (CFA®) designation is fast becoming a requirement for serious investment professionals. Although it requires a successful completion of three levels of grueling exams,it also entails promising careers with lucrative salaries. A student of finance is curious about the average salary of a CFA® charterholder. He takes a random sample of 36 recent charterholders and computes a mean salary of $162,000 with a standard deviation of $36,000. Use this sample information to determine the upper bound of the 90% confidence interval for the average salary of a CFA® charterholder. (Round the "t" value to 3 decimal places.)

You work for a large retailer and have been asked to estimate the proportion of your customers that are less than 30 years old. You have sampled a large number of stores and have found that of the 106 customers you have surveyed, 58 are less than 30 years old.

Assuming your sample is valid, what is the upper bound of a 99% confidence interval for the population proportion of customers who are less than 30 years old?
(Report your answer as a decimal and not as a percentage. For example, report 0.05 rather than 5%.)

1) For the following data on Year-end Audit times (in days), 17, 20, 25, 27, 19, 19, 20, 32, 26, 23, 24, 23, 27, 38, 21, 23, 22, 28, 33, 18, 27, 20, 23, 27, 31 Prepare a table showing in columns Audit Time Intervals (days), Frequencies, Cumulative Frequencies, Relative Frequencies, Cumulative Relative Frequencies, Percent Frequencies, and Cumulative Percent Frequencies.

Researchers selected a simple random sample of 4048 medical records of adults diagnosed with gum disease. In all, 2226 were current smokers, 891 were former smokers, and 931 never smoked regularly. Their research question is:

Do these data indicate that gum disease is equally likely regardless of smoking status?

Using a significance level of 0.05, what is the appropriate conclusion for this test?

The data are consistent with an equal representation of current, former, and never smokers among adults diagnosed with gum disease.

Current smokers make up a significantly greater proportion of adults diagnosed with gum disease than former or never smokers.

Current smokers are most likely to have gum disease.

There is significant evidence that current, former, and never smokers are not equally represented among adults diagnosed with gum disease.

In automobile mileage and gasoline-consumption testing, 6 automobiles were road tested for 300 miles in both city and highway driving conditions. The following data were recorded for miles-per-gallon performance. City 16.2 16.7 15.9 14.4 16 16.2 Highway 19.4 20.6 18.3 18.6 18.6 18.7 Use mean, median, and mode to make a statement about the difference in performance for city and highway driving. Which area of Statistics helps you to either validate or disprove such a statement and why?