(1)The following two claims are similar to the claim in the triangle problem discussed in lecture, but there are subtle differences. Either prove or disprove each claim.
(a) Let T(n) be: C(n, 3) triangles are formed by n lines in the plane if no three of the lines intersect at a single point. ∀n ∈ N, n ≥ 3, T(n).
(b) Let R(n) be: C(n, 3) triangles are formed by n non-parallel lines in the plane. ∀n ∈ N, n ≥ 3, R(n).
In: Math
Create a Normally (Gaussian) distributed random variable1 X with a mean µ and standard deviation σ.
• [20] Create normally distributed 50 samples (Y) with µ and σ, and plot the samples.
• [20] Create normally distributed 5000 samples (X) with µ and σ, and (over) plot the samples.
• [20] Plot the histogram of random variable X and Y. Do not forget to normalize the histogram.
• [35] Plot the Gaussian PDF and its CDF function over the histogram of random variables Y and X.
Do not forget, interpreting the results is the key to properly learn!!
In: Math
A random sample is drawn from a normally distributed population with mean μ = 18 and standard deviation σ = 2.3. [You may find it useful to reference the z table.]
b. Calculate the probabilities that the sample mean is less than 18.6 for both sample sizes. (Round intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)
|
In: Math
If you were the manager of a big company, please provide a scenario/or an application that you would use one or combinations of these two techniques( choose from descriptive statistics, graphs, one sample test, two samples test , ANOVA, two-way ANOVA) to help you in the decision making process.
In: Math
20- The thicknesses of 73randomly selected linoleum tiles were found to have a variance of 3.14 . Construct the 90% confidence interval for the population variance of the thicknesses of all linoleum tiles in this factory. Round your answers to two decimal places.
Step 1 of 1:
In: Math
On an ostrich farm, the weights of the birds are found to be normally distributed. The weights of the females have a mean 78.6 kg and a standard deviation of 5.03 kg. The weights of the males have a mean 91.3 kg and a standard deviation of 6.29 kg. Find the probability that a randomly selected: Male will weight less than 80 kg. Female will weight less than 80 kg. Female will weigh between 70 and 80 kg. 20% of females weigh less than k kg. Find k. The middle 90% of the males weigh between a kg and b kg. Find the values of a and b kg.
In: Math
The following data come from a study designed to investigate drinking problems among college students. In 1983, a group of students were asked whether they had ever driven an automobile while drinking. In 1987, after the legal drinking age was raised, a different group of college students were asked the same question. SHOW EXCEL CODES
Drove While Drinking Year
1983 1987 Total
Yes 1250 991 2241
No 1387 1666 3053
Total 2637 2657 5294
A. Use the chi-square test to evaluate the null hypothesis that population proportions of students who drove while drinking are the same in the two calendar years.
B. What do you conclude about the behavior of college students?
C. Again test the null hypothesis that the proportions of students who drove while drinking are identical for the two calendar years. This time, use the method based on the normal approximation to the binomial distribution that was presenting in Section 14.6. Do you reach the same conclusion?
D. Construct a 95% confidence interval for the true difference in population proportions.
E. Does the 95% confidence interval contain the value 0? Would you have expected it to?
In: Math
Since an instant replay system for tennis was introduced at a major tournament, men challenged
14371437
referee calls, with the result that
431431
of the calls were overturned. Women challenged
745745
referee calls, and
227227
of the calls were overturned. Use a
0.010.01
significance level to test the claim that men and women have equal success in challenging calls. Complete parts (a) through (c) below.
a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of male tennis players who challenged referee calls and the second sample to be the sample of female tennis players who challenged referee calls. What are the null and alternative hypotheses for the hypothesis test?
A.
Upper H 0H0:
p 1p1not equals≠p 2p2
Upper H 1H1:
p 1p1equals=p 2p2
B.
Upper H 0H0:
p 1p1equals=p 2p2
Upper H 1H1:
p 1p1greater than>p 2p2
C.
Upper H 0H0:
p 1p1less than or equals≤p 2p2
Upper H 1H1:
p 1p1not equals≠p 2p2
D.
Upper H 0H0:
p 1p1equals=p 2p2
Upper H 1H1:
p 1p1less than<p 2p2
E.
Upper H 0H0:
p 1p1equals=p 2p2
Upper H 1H1:
p 1p1not equals≠p 2p2
F.
Upper H 0H0:
p 1p1greater than or equals≥p 2p2
Upper H 1H1:
p 1p1not equals≠p 2p2
Identify the test statistic.
zequals=negative . 23−.23
(Round to two decimal places as needed.)
Identify the P-value.
P-valueequals=. 818.818
(Round to three decimal places as needed.)
What is the conclusion based on the hypothesis test?
The P-value is
greater than
the significance level of
alphaαequals=0.010.01,
so
fail to reject
the null hypothesis. There
is not sufficient
evidence to warrant rejection of the claim that women and men have equal success in challenging calls.
b. Test the claim by constructing an appropriate confidence interval.
The
9999%
confidence interval is
nothingless than<left parenthesis p 1 minus p 2 right parenthesisp1−p2less than<nothing.
(Round to three decimal places as needed.)
What is the conclusion based on the confidence interval?
Because the confidence interval limits
▼
do not include
include
0, there
▼
does
does not
appear to be a significant difference between the two proportions. There
▼
is not sufficient
is sufficient
evidence to warrant rejection of the claim that men and women have equal success in challenging calls.
c. Based on the results, does it appear that men and women may have equal success in challenging calls?
A.
The confidence interval suggests that there is a significant difference between the success of men and women in challenging calls. It is reasonable to speculate that women have more success.
B.
The confidence interval suggests that there is no significant difference between the success of men and women in challenging calls.
C.
The confidence interval suggests that there is a significant difference between the success of men and women in challenging calls. It is reasonable to speculate that men have more success.
D.
There is not enough information to reach a conclusion.
In: Math
A bus travels between two cities A and B that are 100 miles apart.Two service stations are located at mile 30 and mile 70, as well as in the cities themselves. The bus breaks down on the road. Assuming the place of breakdown is uniformly distributed between the cities, what is the probability that it is no more than 10 miles to the nearest service station? What is the expectation of the distance to the nearest service station?
In: Math
The researchers classified gas turbines into three categories: traditional, advanced, and aeroderivative. Mean heat rate and standard deviation of heat rate for
Is there sufficient evidence of a difference between the mean heat rates of traditional turbines and aeroderivative turbines at alpha =0.05 ? Show all the work
In: Math
Briefly describe the product-process matrix and the customer-contact matrix for service processes。
In: Math
In a clinical trial, 401,974 adults were randomly assigned to two groups. The treatment group consisted of 201,229 adults given a vaccine and the other 200,745 adults were given a placebo. Among the adults in the treatment group, 33 adults developed the disease and among the placebo group, 115 adults developed the disease. The doctors' claim that the rate for the group receiving the vaccine is less than the group receiving the placebo. Answer the following questions:
a. If w idenitfy the symbolic null and alternative hypothesis.
b. If the P-value for this test is reported as "less than 0.001", what is your decision? What would you conclude about the original claim?
c. Assume that we want to use a 0.05 significance level to test the claim that p1 < p2. If we want to test that claim by using a confidence interval, what confidence level should we use?
d. If we test the original claim, we get the confidence interval -0.000508 < p1 - p2 < −0.000309 , what does this confidence interval suggest about the claim? e. In general, when dealing with inferences for two population proportions, which two of the following are equivalent: Confidence Interval method; P-value method; Critical Value method? Explain.
In: Math
Jobs and productivity! How do retail stores rate? One way to answer this question is to examine annual profits per employee. The following data give annual profits per employee (in units of 1 thousand dollars per employee) for companies in retail sales. Assume σ ≈ 3.7 thousand dollars.
|
4.2 |
6.6 |
4.0 |
8.7 |
7.5 |
6.0 |
8.2 |
5.8 |
2.6 |
2.9 |
8.1 |
−1.9 |
|
11.9 |
8.2 |
6.4 |
4.7 |
5.5 |
4.8 |
3.0 |
4.3 |
−6.0 |
1.5 |
2.9 |
4.8 |
|
−1.7 |
9.4 |
5.5 |
5.8 |
4.7 |
6.2 |
15.0 |
4.1 |
3.7 |
5.1 |
4.2 |
(a) Use a calculator or appropriate computer software to find
x for the preceding data. (Round your answer to two
decimal places.)
thousand dollars per employee
(b) Let us say that the preceding data are representative of the
entire sector of retail sales companies. Find an 80% confidence
interval for μ, the average annual profit per employee for
retail sales. (Round your answers to two decimal places.)
| lower limit | thousand dollars |
| upper limit | thousand dollars |
c) Find an 95% confidence interval for μ, the average annual profit per employee for retail sales. (Round your answers to two decimal places.)
| lower limit | thousand dollars |
| upper limit | thousand dollars |
In: Math
The following data represent petal lengths (in cm) for independent random samples of two species of Iris.
Petal length (in cm) of Iris virginica: x1; n1 = 35
| 5.3 | 5.6 | 6.3 | 6.1 | 5.1 | 5.5 | 5.3 | 5.5 | 6.9 | 5.0 | 4.9 | 6.0 | 4.8 | 6.1 | 5.6 | 5.1 |
| 5.6 | 4.8 | 5.4 | 5.1 | 5.1 | 5.9 | 5.2 | 5.7 | 5.4 | 4.5 | 6.4 | 5.3 | 5.5 | 6.7 | 5.7 | 4.9 |
| 4.8 | 5.9 | 5.1 |
Petal length (in cm) of Iris setosa: x2; n2 = 38
| 1.4 | 1.6 | 1.4 | 1.5 | 1.5 | 1.6 | 1.4 | 1.1 | 1.2 | 1.4 | 1.7 | 1.0 | 1.7 | 1.9 | 1.6 | 1.4 |
| 1.5 | 1.4 | 1.2 | 1.3 | 1.5 | 1.3 | 1.6 | 1.9 | 1.4 | 1.6 | 1.5 | 1.4 | 1.6 | 1.2 | 1.9 | 1.5 |
| 1.6 | 1.4 | 1.3 | 1.7 | 1.5 | 1.6 |
(a) Use a calculator with mean and standard deviation keys to calculate x1, s1, x2, and s2. (Round your answers to two decimal places.)
| x1 = | |
| s1 = | |
| x2 = | |
| s2 = |
(b) Let μ1 be the population mean for
x1 and let μ2 be the
population mean for x2. Find a 99% confidence
interval for μ1 − μ2.
(Round your answers to two decimal places.)
| lower limit | |
| upper limit |
In: Math
David E. Brown is an expert in wildlife conservation. In his book The Wolf in the Southwest: The Making of an Endangered Species (University of Arizona Press), he records the following weights of adult grey wolves from two regions in Old Mexico.
Chihuahua region: x1 variable in pounds
| 86 | 75 | 91 | 70 | 79 |
| 80 | 68 | 71 | 74 | 64 |
Durango region: x2 variable in pounds
| 68 | 72 | 79 | 68 | 77 | 89 | 62 | 55 | 68 |
| 68 | 59 | 63 | 66 | 58 | 54 | 71 | 59 | 67 |
(a) Use a calculator with mean and standard deviation keys to calculate x1, s1, x2, and s2. (Use 2 decimal places.)
| x1 | |
| s1 | |
| x2 | |
| s2 |
(b) Let μ1 be the mean weight of the population
of all grey wolves in the Chihuahua region. Let
μ2 be the mean weight of the population of all
grey wolves in the Durango region. Find a 99% confidence interval
for μ1 – μ2. (Use 2 decimal
places.)
| lower limit | |
| upper limit |
In: Math