The manufacturer of a sports car claims that the fuel injection system lasts 48 months before it needs to be replaced. A consumer group tests this claim by surveying a random sample of 10 owners who had the fuel injection system replaced. The ages of the cars at the time of replacement were (in months):

(i) Use your calculator to calculate the mean age of a car when the fuel injection system fails x and standard deviation s. (Round your answers to two decimal places.)

(ii) Test the claim that the fuel injection system lasts less than an average of 48 months before needing replacement. Use a 5% level of significance.

What are we testing in this problem?

single proportionsingle mean    

(a) What is the level of significance?

The Nero Match Company sells matchboxes that are supposed to have an average of 40 matches per box, with σ = 8. A random sample of 98 matchboxes shows the average number of matches per box to be 42.4. Using a 1% level of significance, can you say that the average number of matches per box is more than 40?

What are we testing in this problem?

single mean single proportion    

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 40; H₁: p < 40H₀: μ = 40; H₁: μ ≠ 40    H₀: p = 40; H₁: p ≠ 40H₀: μ = 40; H₁: μ < 40H₀: p = 40; H₁: p > 40H₀: μ = 40; H₁: μ > 40

(b) What sampling distribution will you use? What assumptions are you making?

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

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that the average number of matches per box is now greater than 40.There is insufficient evidence at the 0.01 level to conclude that the average number of matches per box is now greater than 40.    

1. In a study of children with a particular disorder, parents were asked to rate their child on a variety of items related to how well their child performs different tasks. One item was "Has difficulty organizing work," rated on a five-point scale of 0 to 4 with 0 corresponding to "not at all" and 4 corresponding to "very much." The mean rating for 274 boys with the disorder was reported as 2.34 with a standard deviation of 1.07. (Round your answers to four decimal places.)

Compute the 90%, 95%, and 99% confidence interval

22.)

We never conclude that null hypothesis is true.

a) True

b) False

23.)

When we reject a null hypothesis, we are certain that the alternative hypothesis is true.

a) True

b) False

24)

If a null hypothesis is rejected with a significance level of 0.05, it is also rejected with a significance level of 0.01?

a) True

b) False

c) it might be either

25)

If a null hypothesis is rejected with a significance level of 0.01, it is also rejected with a significance level of 0.05?

a) True

b) False

c) it might be either

PART 1

The height of women on the basketball team is argued to be 5.5 feet tall. Test the data set of Women’s heights and determine if there is a statistically significant difference between the data set and the test value of 5.5. Perform a 2-sided test and use a significance level of 0.05. State your hypotheses Ho and Ha, the test statistic, the p-value and your conclusion. Also, based on your conclusion, what type of error (Type I or Type II) might have you committed? What is the probability associated with the type of error you chose?

PART 2

The height of men on the basketball team is argued to be 6.1 feet tall. Test the data set of Men’s heights and determine if there is a statistically significant difference between the data set and the test value of 6.1. Perform a 2-sided test and use a significance level of 0.05. State your hypotheses Ho and Ha, the test statistic, the p-value and your conclusion. Also, based on your conclusion, what type of error (Type I or Type II) might have you committed? What is the probability associated with the type of error you chose?

PART 3

Calculate 95% confidence intervals for the true mean height of Women and true men height of Men. Interpret your confidence intervals. Do they overlap. Based on whether the confidence intervals overlap or not, what does this say about whether the true mean height of men could equal the true mean height of women?

Suppose that a certain brand of calculators claims to have a 7% defect rate.
a. If we select 6 calculators at random, what is the probability that two or fewer of them are defective? Round your answer to three decimal places.

ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ

b. A researcher believes that the defect rate is actually higher than what the company claims. She selects 500 calculators and finds that 41 of them are defective. At a significance level of 10%, does the researcher have evidence to support her claim? Perform the appropriate hypothesis test. Round your sample statistic to three decimal places.
0.
1.
2.
3.
4.
5.

The paper “Effect of Temperature on the pH of Skim Milk” (Journal of Dairy Research) reported on a study involving x = temperature (degrees ^oC) under specified experimental conditions and y = milk pH. The following data is a representative subset of the data used in the study.

X         14        18        23        24        26        38        38        40

Y         6.89     6.72     6.63     6.64     6.75     6.66     6.57     6.58

X         45        50        55        56        60        67        70        78

Y         6.59     6.48     6.42     6.41     6.38     6.34     6.32     6.34

a. Graph a scatter plot of X vs. Y. Does this data suggest a linear relationship between temperature and pH? Explain why or why not.

b. Show the estimated linear regression line on the scatter plot in (1.).

c. Calculate the sample correlation between x and y. Would you consider the correlation weak, moderate, or strong?

d. For a give temperature = 20 degrees ^oC, calculate the estimated value of Y using the regression line equation

A cell phone company states that the mean cell phone bill of all their customers is less than $83. A sample of 19 customers gives a sample mean bill of $82.17 and a sample standard deviation of $2.37. At ? = 0.05 , test the company’s claim?

1). State the hypothesis and label which represents the claim: : H 0 : H a

2). Specify the level of significance  =

3). Sketch the appropriate distribution, find and label the Critical Value(s) in Statdisk to 3 decimal places, and shade in  to indicate the Rejection Region(s)

4.) Calculate the test statistic (to 3 decimal places). Label this appropriately as Z or t.

5). Decision: Reject Ho or Do Not Reject Ho (You do not have to write a statement):

6). In the distribution below, label the test statistic you calculated from part (4). Using Statdisk, find the P-value corresponding to the test statistic. Shade in and label the P-value in the distribution.

A. A researcher was interested in comparing the GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 13 students from college B yielded the following results. The mean GPA for college A was x1 = 3.11, with a standard deviation s1 = 0.44. The GPA for college B was 2 = 3.44, with a standard deviation s2 = 0.55. Determine a 95% confidence interval for the difference, µ1-µ2 between the mean GPA of college A students and the mean GPA of college B students. (Assume college a and B have the same population standard deviations)

B. A manufacturing process produces bags of cookies. The distribution of content weights of these bags is Normal with mean 15.0 oz and standard deviation 6.0 oz. We will randomly select a sample of 900 bags of cookies and weigh the contents of each bag selected, is the mean of such sample

a) The mean of is x

b) The standard deviation of x is

c) The distribution of x is

d) Does the distribution of depend on the assumption that the weight (in oz) of these bags is normally distributed and why?

C.Last year, the mean annual salary for adults in one town was $35,000. A researcher wants to perform a hypothesis test to determine whether the mean annual salary for adults in this town has changed this year. The mean annual salary for a random sample of 16 adults from the town was 33000. Assume population standard deviation σ =12000. Use a significance level of α=0.05

D. A special diet is intended to reduce the cholesterol of patients at risk of heart disease. After six months on the diet, an SRS of 64 patients at risk for heart disease had an average cholesterol of x= 192, with standard deviation s = 24. The 95% confidence interval for the average cholesterol of patients at risk for heart disease who have been on the diet for 6 months is

USE R AND SHOW CODES!!

1.a. An investigator is interested in comparing the cardiovascular fitness of elite runners on three different training courses. course one is at, course 2 has graded inclines, and

course three includes steep inclines, Ten runners were involved for each course. Heart rates measured on each course are as the following table

Course 1 Course 2 Course 3

132 135 138

143 148 148

135 138 141

128 131 139

141 141 150

150 156 161

131 134 138

150 156 162

142 145 151

139 165 160

Is there a significant difference in the mean heart rates of runners on three courses? alpha= 0:05

1.b. The following data is collected on the enzyme activity of MPI (mannose-6-phosphate isomerase) and MPI genotypes separated for male and female.

a. Is there any significant difference between male and female?

b. Is there any significant difference between genotypes?

c. Is there any interaction between sex and genotypes?

DATA

Genotype Female Male

FF 2.838 1.884

4.216 2.889

4.198 2.283

4.939 3.486

FS 3.55 2.396

4.556 2.956

3.087 3.105

1.943 2.649

SS 3.620 2.801

3.079 3.421

3.586 4.275

2.669 3.110

explain some distinguishing feature of non parametric tests?

Suppose that your favorite restaurant serves the best ramen dishes in town . Since you eat there so often, you've observed the following :

the patrons are 40% male ,

the probability that the patron orders a tofu-based meal given that the patron is male is 30%,

while the probability that the order is tofu-based given the patron is female is 50%.

One night, you walk into the restaurant and see a tofu-based dish on one of the tables that is partially eaten ,but the customer has already left. What is the probability that the customer was female?

A bag of marbles contains 3 blue marbles, 2 green marbles. 2 marbles are selected from the bag without replacement. What is the probability of getting 1 green marble?

The random variable Y has an exponential distribution with probability density function (pdf)

as follows:

f(y) = λe−λy, y >0

= 0, otherwise

(i) Showing your workings, find P (Y > s|Y > t), for s ≥ t. [3]

(ii) Derive an expression for the conditional pdf of Y , conditional on that Y ≤ 200. [3]

N(t) is a Poisson process with rate λ

(iii) Find an expression for the Cumulative Distribution Function (CDF) of the waiting time until the first event. (Hint: Consider the probability of there being 0 events in time t.) [2]

(iv) Explain the relationship between the mean of the Poisson distribution with rate λ and the mean of the associated distribution for the waiting time.

**Assume that both Plain and Peanut M&M candies are normally distributed.

1. Suppose that M&M claims that their Plain M&Ms have an equal proportion of red and brown.

a. Test the claim that the proportion of red M&Ms is greater than the proportion of brown M&Ms.

b. Test the claim that the proportion of red M&Ms is less than the proportion of brown M&Ms.

Total number of M&M's: 665

Total number of red M&M's: 76

Total number of brown M&M's: 66