A) Many celebrities and public figures have Twitter accounts with large numbers of followers. However, some of these followers are fake, resulting from accounts generated by spamming computers. In a sample of 46 twitter audits, the mean percentage of fake followers was ?̅= 15.9 with a standard deviation of ? = 9.3. Construct a 99% confidence interval for the mean percentage of fake Twitter followers.

B) An executive at Twitter claims that the percentage of fake Twitter followers is less than 10. Based on the confidence interval, is this a reasonable claim?

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

Method 1: Use the Student's t distribution with d.f. = n − 1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 41, with sample mean x = 45.7 and sample standard deviation s = 6.4.

(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(b) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(c) Now consider a sample size of 71. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(d) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

Do online students perform differently than students in a traditional classroom? Last semester there were 100 students registered for the online version of a statistics course, and 100 students registered for the traditional classroom. All students took the same final exam. Suppose the group of online students had a mean exam score of 75 with a standard deviation of 3. The classroom students had a mean score of 76 with a standard deviation of 4. Does the sample data provide evidence that there is a difference in the average exam scores between the two groups? Test the relevant hypotheses at a significant level of x = 0.05.

Show your work.

Thirty-two small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 44.9 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

As the confidence level increases, the margin of error remains the same.As the confidence level increases, the margin of error decreases.     As the confidence level increases, the margin of error increases.

(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

As the confidence level increases, the confidence interval increases in length.As the confidence level increases, the confidence interval remains the same length.     As the confidence level increases, the confidence interval decreases in length.

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 15 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.22 gram.

When finding an 80% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)

z_c =

(a)

Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limitupper limitmargin of error

(b)

What conditions are necessary for your calculations? (Select all that apply.)

normal distribution of weightsuniform distribution of weightsσ is knownσ is unknownn is large

(c)

Interpret your results in the context of this problem.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.     There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

(d)

Which equation is used to find the sample size n for estimating μ when σ is known?

n =

n =

     n =

n =

Find the sample size necessary for an 80% confidence level with a maximal margin of error  E = 0.08 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

hummingbirds

Suppose that Randy is an analyst for the bicyling industry and wants to estimate the asking price of used entry-level road bikes advertised online in the southeastern part of the United States. He obtains a random sample of n = 14 online advertisements of entry-level road bikes. He determines that the mean price for these 14 bikes is ¯¯¯ x = $ 714.19 and that the sample standard deviation is s = $ 184.56 . He uses this information to construct a 99% confidence interval for μ , the mean price of a used road bike. What is the lower limit of this confidence interval? Please give your answer to the nearest cent.

Every semester, I would like for more than 75% of my students to score a 70 or higher on the first test. This semester, out of the 72 students who took the first test, 59 got at least a C (scored higher a 70 or higher). Is there sufficient evidence to conclude, at the 10% significance level, that more than 75% of the students got at least a C on the first exam? Find the p-value.

Identify the null and alternative hypotheses, test statistic, critical value(s) and critical region or p-value, as indicated, and state the final conclusion that addresses the problem. Show all seven steps.

The mean incubation time for a type of fertilized egg kept at a certain temperature is 19 days. Suppose that the incubation times are approximately normally distributed with a standard deviation of 1 day. Complete parts​ (a) through​ (e) below.

(a) Draw a normal model that describes egg incubation times of these fertilized eggs.

(b)Find and interpret the probability that a randomly selected fertilized egg hatches in less than

17 days.

(c) Find and interpret the probability that a randomly selected fertilized egg takes over 21 days to hatch.

(d) Find and interpret the probability that a randomly selected fertilized egg hatches between

15 and 19 days.

(e) Would it be unusual for an egg to hatch in less than 13 days? Why?

A poll is taken in which 348 out of 525 randomly selected voters indicated their preference for a certain candidate.

(a) Find a 99% confidence interval for p.

≤. p. ≤.

(b) Find the margin of error for this 99% confidence interval for p.

For which combination of population and sample size listed below will you find the sampling distribution of the sample mean approximately normally distributed?

a) Population is Right Skewed and n = 10

b) Population is Right Skewed and n = 40

c) Population is Bell Shaped and n = 10

d) B and C only

e) A, B and C

•An automobile insurer has found that repair claims have a mean of $920 and a standard deviation of $870. Suppose that the next 100 claims can be regarded as a random sample from the long-run claims process.

What is the probability that the average of the 100 claims is larger than $900?

In this assignment students will demonstrate their understanding of the distribution of means doing all steps of hypothesis testing.
For each problem students will write out all steps of hypothesis testing including populations, hypotheses, cutoff scores, and all relevant calculations. Assignments will be typed and uploaded in a word document to blackboard.

1.A nationwide survey in 1995 revealed that U.S. grade-school children spend an average of µ = 8.4 hours per week doing homework. The distribution is normal with σ = 3.2. Last year, a sample of n = 100 grade-school children was given the same survey. For this sample, the mean number of homework hours was 7.1. Has there been a significant change in the homework habits of grade-school children? Test with α = .05.

2.On the basis of her newly developed technique, a student believes she can reduce the amount of time schizophrenics spend in an institution. As director of training at a nearby institution, you agree to let her try her method on 20 schizophrenics, randomly sampled from your institution. The mean duration that schizophrenics stay at your institution is 85 weeks, with a standard deviation of 15 weeks. The scores are normally distributed. The results of the experiment show that patients treated by the student stay at the institution a mean duration of 78 weeks. What do you conclude about the student’s technique? Use α = .05.

3.A psychologist has developed a standardized test for measuring the vocabulary skills of 4-year-old children. The scores on the test form a normal distribution with μ = 60 and σ = 10. A researcher would like to use this test to investigate the idea that children who grow up with no siblings develop vocabulary skills at a different rate than children in large families. A sample of n = 25 children is obtained, and the mean test score for this sample is 63. On the basis of this sample, can the researcher conclude that vocabulary skills for children with no siblings are significantly different from those of the general population? Test at the .01 level of significance.

4.The average age for licensed drivers in a county is 42.6, with a standard deviation of 12, and the distribution is approximately normal. A county police officer was interested in whether the average age of those receiving speeding tickets is less that the average age of the population who has a license. She obtained a sample of 16 drivers with speeding tickets. The average age for this sample was 34.4. Do all the steps of hypothesis testing using the 0.01 significance level.

A simple random sample with n=54 provided a sample mean of 24 and a sample standard deviation of 4.3 .

a. Develop a 90% confidence interval for the population mean (to 1 decimal).

b. Develop a 95% confidence interval for the population mean (to 1 decimal).

c. Develop a 99% confidence interval for the population mean (to 1 decimal).

What happens to the margin of error and the confidence interval as the confidence level is increased?

According to the National Automobile Dealers Association, the mean price for used cars is $10,192. A manager of a Kansas City used car dealership reviewed a sample of 50 recent used car sales at the dealership in an attempt to determine whether the population mean price for used cars at this particular dealership differed from the national mean. The prices for the sample of 50 cars are shown.

1. Use Excel’s Data Analysis to get the descriptive statistics for sale price.
2. Construct a 95% confidence interval for the population mean sale price for the dealership.
3. Formulate the hypotheses that can be used to determine whether the population mean price for used cars at this particular dealership differed from the national mean. Conduct the test using the critical value approach and α=0.05.
4. Use Excel function =t.dist.2t to find the p-value in the preceding hypothesis test and construct the test with the p-value approach.
5. Formulate hypotheses that can be used to determine whether the population mean price for used cars at this particular dealership is less than the national mean. At α=0.05, can your null hypothesis be rejected? What is your conclusion? Conduct the test using the critical value approach.
6. Use Excel function =t.dist to find the p-value in the preceding hypothesis test and construct the test with the p-value approach.

The length of time, in hours, it takes an "over 40" group of people to play one soccer match is normally distributed with a mean of 4 hours and a standard deviation of 0.5 hours. A sample of size n = 50 is drawn randomly from the population. Find the probability that the sample mean is between 3.5 hours and 4.1 hours.