When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ. (Notice that, When σ is unknown and the sample is of size n < 30, there is only one method for constructing a confidence interval for the mean by using the Student's t distribution with d.f. = n - 1.) 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 = 30, with sample mean x = 45.2 and sample standard deviation s = 5.3.

(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.

90% Lower limit and Upper limit

95% Lower limit and Upper limit

99% Lower limit and Upper limit

(d) Now consider a sample size of 50. 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.

90% Lower limit and Upper limit

95% Lower limit and Upper limit

99% Lower limit Upper limit

Please show me how to do this on a TI 84 Calculator if possible. Thank you!

The Normal Probability distribution has many practical uses. Please provide some examples of real life data sets that are normally distributed.

Researchers gave 40 index cards to a waitress at an Italian restaurant in New Jersey. Before delivering the bill to each customer, the waitress randomly selected a card and wrote on the bill the same message that was printed on the index card. Twenty of the cards had the message "The weather is supposed to be really good tomorrow. I hope you enjoy the day!" Another 20 cards contained the message "The weather is supposed to be not so good tomorrow. I hope you enjoy the day anyway!"

After the customers left, the waitress recorded the amount of the tip, percent of bill, before taxes. Given are the tips for those receiving the good‑weather message.

Given are the tips for the 20 customers who received the bad‑weather message.

Stemplots for both data sets are shown.

Neither stemplot suggests a strong skew or the presence of strong outliers. Because of this, t procedures are reasonable here.

Is there good evidence that the two different messages produce different percent tips?

Let μ1 be the mean tip percent when the forecast is good, and let μ2 be the mean tip percent when the forecast is bad. Select the correct hypotheses statements that we want to test.

H0:μ1=μ2 versus Ha:μ1>μ2

H0:μ1=μ2 versus Ha:μ1≠μ2

H0:μ1=μ2 versus Ha:μ1<μ2

H0:μ1≠μ2 versus Ha:μ1<μ2

What degrees of freedom (df) would you use in the conservative two‑sample t procedures to compare the percentage of tips when the forecast is good and bad? (Enter your answer as a whole number.)

df=

What is the two‑sample t test statistic (rounded to three decimal places)?

t=

Test whether there is good evidence that the two different messages produce different percent tips at α=0.1 . The null hypothesis of no difference in tips due to the weather "forecast" is

not rejected.

rejected.

The following data represent crime rates per 1000 population for a random sample of 46 Denver neighborhoods.†

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to one decimal place.)

(b) Let us say the preceding data are representative of the population crime rates in Denver neighborhoods. Compute an 80% confidence interval for μ, the population mean crime rate for all Denver neighborhoods. (Round your answers to one decimal place.)

(c) Suppose you are advising the police department about police patrol assignments. One neighborhood has a crime rate of 61 crimes per 1000 population. Do you think that this rate is below the average population crime rate and that fewer patrols could safely be assigned to this neighborhood? Use the confidence interval to justify your answer.

Yes. The confidence interval indicates that this crime rate is below the average population crime rate.

Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.     

No. The confidence interval indicates that this crime rate is below the average population crime rate.

No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(d) Another neighborhood has a crime rate of 75 crimes per 1000 population. Does this crime rate seem to be higher than the population average? Would you recommend assigning more patrols to this neighborhood? Use the confidence interval to justify your answer.

Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate.     

No. The confidence interval indicates that this crime rate is higher than the average population crime rate.

No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(e) Compute a 95% confidence interval for μ, the population mean crime rate for all Denver neighborhoods. (Round your answers to one decimal place.)

(f) Suppose you are advising the police department about police patrol assignments. One neighborhood has a crime rate of 61 crimes per 1000 population. Do you think that this rate is below the average population crime rate and that fewer patrols could safely be assigned to this neighborhood? Use the confidence interval to justify your answer.

Yes. The confidence interval indicates that this crime rate is below the average population crime rate.

Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.   

No. The confidence interval indicates that this crime rate is below the average population crime rate.

No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(g) Another neighborhood has a crime rate of 75 crimes per 1000 population. Does this crime rate seem to be higher than the population average? Would you recommend assigning more patrols to this neighborhood? Use the confidence interval to justify your answer.

Yes. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

Yes. The confidence interval indicates that this crime rate is higher than the average population crime rate.     

No. The confidence interval indicates that this crime rate is higher than the average population crime rate.

No. The confidence interval indicates that this crime rate does not differ from the average population crime rate.

(h) In previous problems, we assumed the x distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: Use the central limit theorem.

Yes. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

Yes. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.     

No. According to the central limit theorem, when n ≥ 30, the x distribution is approximately normal.

No. According to the central limit theorem, when n ≤ 30, the x distribution is approximately normal.

2. The IQ of humans is approximately normally distributed with a mean 100 and standard deviation of 15. A. What is the probablitlty that a randomly selected person has an IQ greater than 105? B. What is the probablitlty that a SPS of 60 randomly selected people will have a mean IQ greater than 105?

3. A 95% confidence interval for a population mean is (57,65). Can you reject the null hypothesis the mean= 68 at the 5% significance level why or why not?

2. A researcher wishes to determine whether there is a difference in the average age of elementary school, high school, and community college teachers. Teachers are randomly selected from each group. Their ages are recorded below. Test the claim that at least one mean is different from the others. Use α = 0.01.
Resource: The One-Way ANOVA

A pollster surveyed a sample of 980 adult Americans, asking them if they own a personal firearm. 34% of the sample said yes.

1. What is a 90% confidence interval estimate for the percentage of Americans that own a firearm?

2. A gun owners’ group claims that more Americans own a firearm in 2015 than ten years ago, when the percentage of owners was 30%. At the 0.05 level of significance, has the percentage of owners increased?

3. What is the p-value for this problem? How does it tell you what conclusion to draw about the null hypothesis?

2- You are a researcher studying the affect of happiness between office workers, students, and musicians. Is there a difference in their affect of happiness?

A research center survey of 2 comma 375 adults found that 1 comma 942 had bought something online. Of these online​ shoppers, 1 comma 274 are weekly online shoppers. Complete parts​ (a) through​ (c) below.

1. Construct a​ 95% confidence interval estimate of the population proportion of adults who had bought something online.
2. Construct a​ 95% confidence interval estimate of the population proportion of online shoppers who are weekly online shoppers.
3. How would the director of​ e-commerce sales for a company compare the results of​ (a) and​ (b)?

The therm dataset contains information on survey respondents’ opinions about various public figures. These are “feeling thermometer” scores, which range from 0 (total dislike of the person) to 100 (total like). The relevant variables for this question are:

• white: a dummy variable indicating whether the respondent is white (ie, 1 for white and 0 for non-white)

• ideology: the respondent’s ideology on a scale of 1 (most liberal) to 7 (most conservative)

• obama: the respondent’s “feeling thermometer” score for Barack Obama

(a) One regression with obama as the dependent variable and white and ideology as the independent variables shows that

obama = 111.2 − 21.2 ∗ white − 8.7 ∗ ideology

What is the estimated intercept for white respondents? What about for non-white respondents?

(b) Another regression that includes a term for the interaction between white and ideology shows that

obama = 90.3+ 8.3∗white−3.0∗ideology−7.6∗(white ·ideology)

What are the estimated intercept and slope of ideology among white respondents? What about among non-white respondents? Is the relationship between ideology and Obama opinion stronger for white respondents, or non-white respondents?

1. An ecologist is interested in studying the presence of different types of animal species in different locations. Using the following contingency table and the total sample size, rewrite the frequencies as relative frequencies. Round each relative frequency to two decimal places.

2. You are interested in learning about students' favorite mode of transportation at two universities. Fill in the blanks in the following contingency table, assuming that the variables are independent.

3.Your teacher claims that the final grades in class are distributed as: A, 25%; B, 25%; C, 40%; D, 5%; F, 5%. At the end of a randomly selected academic quarter, the following number of grades are recorded. Calculate the appropriate chi-square test statistic that would be used to determine if the grade distribution for the course is different than expected. Round your answer to two decimal places.

4. A dog breeder wishes to see if prospective dog owners have any preference among six different breeds of dog. A sample of 200 people (prospective dog owners) provided the data below. Find the critical chi-square value that would be used to test the claim that the distribution is uniform. Use α = 0.01 and round your answer to three decimal places.

Calcium levels in people are normally distributed with a mean of 9.7mg/dL and a standard deviation of 0.3mg/dL. Individuals with calcium levels in the bottom 15

% of the population are considered to have low calcium levels. Find the calcium level that is the borderline between low calcium levels and those not considered low. Carry your intermediate computations to at least four decimal places. Round your answer to one decimal place.Calcium levels in people are normally distributed with a mean of

9.7mg/dL and a standard deviation of 0.3mg/dL. Individuals with calcium levels in the bottom 15% of the population are considered to have low calcium levels. Find the calcium level that is the borderline between low calcium levels and those not considered low. Carry your intermediate computations to at least four decimal places. Round your answer to one decimal place.

A recent study found that children who watched a cartoon with food advertising​ ate, on​ average, 28.6 grams of crackers as compared to an average of 18.8 grams of crackers for children who watched a cartoon without food advertising. Suppose that there were 61 children in each​ group, and the sample standard deviation for those children who watched the food ad was 8.7 grams and the sample standard deviation for those children who did not watch the food ad was 7.7 grams. Complete parts​ (a) and​ (b) below.

b. Assuming that the population variances are​ equal, construct 95% confidence interval estimate of the difference μ1−μ2 between the mean amount of crackers eaten by the children who watch and do not watch the food ad.

___≤ μ1−μ2 ≤ ___ (Round to two decimal places as needed.)

Thirty-four 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 41.7 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 decreases.

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

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.