random election of 11 children tested and finds that their mean attention span is 31 minutes with a standard deviation of 8 minutes. assuming attention spans are normally distributed, find a 95% confidence interval for the mean attention span of children.   also calculate the upper and lower limit of the confidence interval

A random sample of n = 1,400 observations from a binomial population produced x = 659.

(a) If your research hypothesis is that p differs from 0.5, what hypotheses should you test?

H₀: p ≠ 0.5 versus H_a: p = 0.5H₀: p = 0.5 versus H_a: p < 0.5     H₀: p = 0.5 versus H_a: p > 0.5H₀: p < 0.5 versus H_a: p > 0.5H₀: p = 0.5 versus H_a: p ≠ 0.5

(b) Calculate the test statistic and its p-value. (Round your test statistic to two decimal places and your p-value to four decimal places.)

Child Health and Development Studies (CHDS) has been collecting data about expectant mothers in Oakland, CA since 1959. One of the measurements taken by CHDS is the age of first time expectant mothers. Suppose that CHDS finds the average age for a first time mother is 26 years old. Suppose also that, in 2015, a random sample of 50 expectant mothers have mean age of 26.5 years old, with a standard deviation of 1.9 years. At the 5% significance level, we conduct a one-sided T-test to see if the mean age in 2015 is significantly greater than 26 years old. Statistical software tells us that the p-value = 0.034.

Which of the following is the most appropriate conclusion?

A) There is a 3.4% chance that a random sample of 50 expectant mothers will have a mean age of 26.5 years old or greater if the mean age for a first time mother is 26 years old. B) There is a 3.4% chance that mean age for all expectant mothers is 26 years old in 2015.

C) There is a 3.4% chance that mean age for all expectant mothers is 26.5 years old in 2015.

D) There is 3.4% chance that the population of expectant mothers will have a mean age of 26.5 years old or greater in 2015 if the mean age for all expectant mothers was 26 years old in 1959.

using chebyshev's inequality, the probability that a random variable will be within 2 standard deviations of its own mean the least?

The distribution of scores on a recent test closely followed a Normal Distribution with a mean of 22 points and a standard deviation of 2 points. For this question, DO NOT apply the standard deviation rule.

(a) What proportion of the students scored at least 25 points on this test, rounded to five decimal places?

(b) What is the 20 percentile of the distribution of test scores, rounded to three decimal places?

A consumer product testing organization uses a survey of readers to obtain customer satisfaction ratings for the nation's largest supermarkets. Each survey respondent is asked to rate a specified supermarket based on a variety of factors such as: quality of products, selection, value, checkout efficiency, service, and store layout. An overall satisfaction score summarizes the rating for each respondent with 100 meaning the respondent is completely satisfied in terms of all factors. Suppose sample data representative of independent samples of two supermarkets' customers are shown below.

(a)

Formulate the null and alternative hypotheses to test whether there is a difference between the population mean customer satisfaction scores for the two retailers. (Let μ₁ = the population mean satisfaction score for Supermarket 1's customers, and let μ₂ = the population mean satisfaction score for Supermarket 2's customers. Enter != for ≠ as needed.)

H₀:

H_a:

(b)

Assume that experience with the satisfaction rating scale indicates that a population standard deviation of 14 is a reasonable assumption for both retailers. Conduct the hypothesis test.

Calculate the test statistic. (Use μ₁ − μ₂. Round your answer to two decimal places.)

Report the p-value. (Round your answer to four decimal places.)

p-value =

At a 0.05 level of significance what is your conclusion?

Reject H₀. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.Do not reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.     Reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.Do not reject H₀. There is not sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.

(c)

Which retailer, if either, appears to have the greater customer satisfaction?

Supermarket 1 Supermarket 2     neither

Provide a 95% confidence interval for the difference between the population mean customer satisfaction scores for the two retailers. (Use x₁ − x₂.Round your answers to two decimal places.)

_______ to ________

use R

# Problem 4 (5 pts each):
# Set x as a vector of 500 random numbers from Unif(100,300).
# This vector will be kept fixed for the rest of this problem.
#
# (a) Define a function b1(x, beta0, beta1, sigm) that uses the lm() function to
# return the regression line slope b1 for y as a linear function of x, where
#
# y = beta0 + beta1 x + err
#
# and the error term 'err' has a normal N(0,sigm^2) distribution
# (note that standard deviation is equal 'sigm').
#
# Hint: See how the slope b1 is extracted in the initial example of Session 11.

# (b) Replicate the function b1 twenty thousand times for
# beta0 = 15, beta1 = 2, and sigm =10, and store into a vector 'Slopes'.

# (c) Plot the empirical density of Slopes.

# (d) Calculate sample mean and sample variance of Slopes.

# (e) Add to the plot the pdf of a Normal distribution with parameters from part (d).

Many hotels have begun a conservation program that encourages guests to re-use towels rather than have them washed on a daily basis. A recent study examined whether one method of encouragement might work better than another. Different signs explaining the conservation program were placed in the bathrooms of the hotel rooms, with random assignment determining which rooms received which sign. One sign mentioned the importance of environmental protection, whereas another sign claimed that 75% of the hotel’s guests choose to participate in the program. The researchers suspected that the latter sign, by appealing to a social norm, would produce a higher proportion of hotel guests who agree to re-use their towels. Researchers used the hotel staff (a mid-sized, mid-priced hotel in the Southwest that was part of a well-known national hotel chain) to record whether guests staying for multiple nights agreed to reuse their towel after the first night.

(a) Identify the observational units, explanatory variable, and response variable in this study.

(b) State the null and alternative hypotheses in symbols, and be sure to define the parameter in the context of this study.

The following table displays the observed data in this study:

(c) Calculate the conditional proportions of re-use in each group.

(e) Use a two-sample z-test to test the hypotheses that you stated in (a). Report the test statistic and p-

value.

(f) Report your test decision at the α = 0.10, 0.05, and 0.01 significance levels. Also summarize what

these test decisions reveal about the strength of evidence for the researchers’ conjecture.

(g) Produce and interpret a 90% confidence interval for the difference in probabilities of re-using towels

between these two signs.

PROBLEM 2.

Based on data from Consumer Reports, replacement times of TVs is on average 3.4 years with the standard deviation of 1.2 years. Answer the following questions.

Question 2 (2 points):

A random sample of 41 TVs is selected. Find the probability that their average replacement time is between 3 and 4 years.

a) 0.6472
b) 0.9827
c) 0.3208
d) None of the above

Question 3 (2 points):

If a random sample of 29 TVs was chosen, what would be the probability that their average replacement time exceeds 3.5 years?

a) 0.6736
b) 0.3264
c) 0.4681
d) None of the above

Listed below are attractiveness ratings made by participants in a speed dating session. Each attribute rating is the sum of the ratings of five attributes (sincerity, intelligence, fun, ambition, shared interests).

Use a 0.05 significance level to test the claim that there is a difference between female attractiveness ratings and male attractiveness rating by following the steps below:

(a) State the null and alternative hypotheses, indicate the significance level and the type of test (left-, right-, or two-tailed test).

(b) Calculate by hand the test statistic.

(c) Use the appropriate sheet in the Hypothesis Test and Confidence Interval template to complete all relevant computations (including the test statistic: compare with (b) to confirm your calculation is correct).

(d) Use the P-value obtained in (c) to explain whether or not the null hypothesis is rejected.

(e) What can be concluded based off this data?

(f) Are there any potential issues related to the validation of the result (Hint: the subjective nature of the measures)

Instructions This assignment is to be typed up in the supplied R-Script. You need to show all of your work in R in the given script.

3. Infant mortality. The infant mortality rate is defined as the number of infant deaths per 1,000 live births. This rate is often used as an indicator of the level of health in a country. The relative frequency histogram below shows the distribution of estimated infant death rates for 224 countries for which such data were available in 2014.

(a) Estimate Q1, the median, and Q3 from the histogram.

(b) Would you expect the mean of this data set to be smaller or larger than the median? Explain your reasoning.

(c) If you calculated the z-score for the median in this distribution, would the result be positive or negative? Explain your reasoning.

The United States Golf Association tests golf balls to ensure that they conform to the rules of
golf. Balls are tested for weight, diameter, roundness and overall distance. The overall distance test is
conducted by hitting balls with a driver swung by a mechanical device nicknamed “Iron Byron” after the
legendary great Byron Nelson, whose swing the machine is said to emulate. Following are 100 distances
(in yards) achieved by a particular brand of golf ball in the overall distance test.

261.3 259.4 265.7 270.6 274.2 261.4 254.5 283.7 258.1 270.5
255.1 268.9 267.4 253.6 234.3 263.2 254.2 270.7 233.7 263.5
244.5 251.8 259.5 257.5 257.7 272.6 253.7 262.2 252.0 280.3
274.9 233.7 237.9 274.0 264.5 244.8 264.0 268.3 272.1 260.2
255.8 260.7 245.5 279.6 237.8 278.5 273.3 263.7 241.4 260.6
280.3 272.7 261.0 260.0 279.3 252.1 244.3 272.2 248.3 278.7
236.0 271.2 279.8 245.6 241.2 251.1 267.0 273.4 247.7 254.8
272.8 270.5 254.4 232.1 271.5 242.9 273.6 256.1 251.6 256.8
273.0 240.8 276.6 264.5 264.5 226.8 255.3 266.6 250.2 255.8
285.3 255.4 240.5 255.0 273.2 251.4 276.1 277.8 266.8 268.5

Construct a frequency distribution for these data using 13 bins. Draw the histogram

When an opinion poll calls landline telephone numbers at random, approximately 30% of the numbers are working residential phone numbers. The remainder are either non-residential, non-working, or computer/fax numbers. You watch the random dialing machine make 20 calls. (Round your answers to four decimal places.)

(a) What is the probability that exactly 4 calls reach working residential numbers?


(b) What is the probability that at most 4 calls reach working residential numbers?


(c) What is the probability that at least 4 calls reach working residential numbers?


(d) What is the probability that fewer than 4 calls reach working residential numbers?


(e) What is the probability that more than 4 calls reach working residential numbers?

A standardized exam consists of three parts: math, writing, and critical reading. Sample data showing the math and writing scores for a sample of 12 students who took the exam follow.

(a)

Use a 0.05 level of significance and test for a difference between the population mean for the math scores and the population mean for the writing scores. (Use math score − writing score.)

Formulate the hypotheses.

H₀: μ_d ≤ 0

H_a: μ_d = 0

H₀: μ_d ≠ 0

H_a: μ_d = 0

H₀: μ_d > 0

H_a: μ_d ≤ 0

H₀: μ_d = 0

H_a: μ_d ≠ 0

H₀: μ_d ≤ 0

H_a: μ_d > 0

Calculate the test statistic. (Round your answer to three decimal places.)

Calculate the p-value. (Round your answer to four decimal places.)

p-value =

What is your conclusion?

Do not reject H₀. We can conclude that there is a significant difference between the population mean scores for the math test and the writing test. Reject H₀. We can conclude that there is a significant difference between the population mean scores for the math test and the writing test.      Do not reject H₀. We cannot conclude that there is a significant difference between the population mean scores for the math test and the writing test. Reject H₀. We cannot conclude that there is a significant difference between the population mean scores for the math test and the writing test.

(b)

What is the point estimate of the difference between the mean scores for the two tests? (Use math score − writing score.)

What are the estimates of the population mean scores for the two tests?

Math:?

Writing:?

Which test reports the higher mean score?

The math test reports a / lower or higher /mean score than the writing test.

Determine the best way to model the relationship between the radon measurement A and B.

3a) graph with your best model represented on it.

3b) What told you it was the best model?

3c) Show your regression model for predicting radon B.