Suppose we want to choose 6 letters, without replacement, from 14 distinct letters.

Q. In this question, you will do some resampling and show results in graphics. This is related to bootstrap technique. The population distribution is a normal with µ = 10 and σ^2 = 4. The statistic is the sample mean. Hence in theory we know exactly what the density function of the sample mean is.

(a) Simulate a sample, say x, with sample size n=100. Report its mean, sd, min, and max.

(b) Use R functions sample and replicate to resample x 50000 times with replacement. The statistic is the sample mean and the output is booted.data. Find the mean, sd, min, and max of booted.data.

(c) Plot the histogram of booted.data. Please double the cells of histogram since the default one is too small. Please plot as a density plot since the theoretical density will be added in the next step. Comment the shape and center of this distribution.

(d) Plot the histogram of booted.data-mean(x) with twice number of default cells. Please plot as a density plot. Add the theoretical density function of the X¯ −µ to the histogram with different line type and color. Comment out your findings.

(e) Repeat the procedures from (a) to (d) two additional times to check consistency.

3. The Associate Dean for Library Services wants to estimate the mean number of times students use the library during a semester. A sample study of 35 students showed a mean of 11 times with a standard deviation of 3 times. A. What is the population mean, using a 99 percent confidence interval? B. What can be concluded from the results? C. What is the population mean, using a 95 percent confidence interval? D. What can be concluded from the results?

13. You wish to test the claim that fewer than 33% of households in a certain city own pets.

• [3 points] Give the null and alternate hypotheses.

• [2 points] Is the test right-tailed, left-tailed or two-tailed?

• The P-value was found to be 0.0673,

Choose using a significance level of 5%:

Reject the null hypothesis or Fail to reject the null hypothesis

Give the conclusion about the claim in an English sentence in the context of the scenario.

A person wants to open a pet supply store in the city, but not if fewer than 33% of the city’s households have pets, Based on your conclusion, should the person open the store?       

1. What is the difference between the “Between Treatment” sources in the One-Way ANOVA compared to the Two-Way ANOVA?

2. Explain how and why the numerators changes when calculating different F-statistics in a Two-Way ANOVA..

3. Explain why the denominator does not change when calculating different F-statistics in a Two-Way ANOVA.

1. A researcher is interested in determining if there is a difference in the amount of time babies pay attention to other babies and to adults.  The researcher takes a group of 26 babies and randomly assigns them to two groups (there are 13 babies in each group).  In one group, babies watch a video of other babies and in the other group, babies watch a video of adults.  The researcher records how long each baby pays attention to the video.  He finds that babies who watched other babies paid attention for 41 seconds with a standard deviation of 5.2 seconds.  On the other hand, babies who watched adults paid attention for 24.6 seconds with a standard deviation of 8.9 seconds.  At a significance level of .01, is there a difference in how long the babies paid attention to each video?
2. State the null and alternative hypotheses (1 pt).
3. Find the critical value(s) (1/2 pt).
4. Compute the observed value of the test statistic (1 pt).
5. Make a decision to reject or fail to reject the null hypothesis at the specified level of significance (1/2 pt).
6. State your conclusion in terms of the alternative hypothesis (research question) (1 pt).
7. Compute the observed value of the test statistic (1 pt).

Researchers are interested in comparing the depression scores (measured on a scale of 1-100) of patients diagnosed with clinical depression before and after they have participated in a new exercise regimen. They take a random sample of 5 people who report low levels of exercise and measure their depression scores. They then have these 5 patients complete a 4-week exercise regimen and measure their depression scores after the 4 weeks. Results of analysis are presented in the table below.

Your research question of interest is:

Is there a difference in the mean depression score of patients before and after they participate in the 4- week exercise regimen?

You are informed the estimated standard error of the mean differences (s) is 1.74

You will use the information above to complete the question parts below for a two related samples t-test.

Part A: Which of the following represents the appropriate null hypothesis (H₀), given this research question of interest?

Part B: Which of the following represents the appropriate alternative hypothesis (H₁), given this research question of interest?

Part C: Which of the following represents the appropriate critical value(s), testing at an alpha level (α) of 0.05?

Part D: What is the t-statistic associated with your test? (rounded to the nearest hundredth) - see note above for the value of the standard error of the mean differences.

Part E: Given your test results, what is your decision about the null hypothesis?

Part F: The best interpretation of the appropriate decision regarding the null hypothesis would be: "Based on our study, we (Have/Have not)

enough evidence to conclude that there appears to be a difference in the population mean depression score before and after a 4 week exercise regimen."

lesson 40 homework study of different in tow population proportions

Sam claims that the proportion of all males who have been locked out of their houses is less than the proportion of females who have been locked out of their houses in the last three years. Sam found that 135 out of 300 randomly selected males had locked themselves out of their houses within the last three years. Also, he found that 220 out of 400 randomly selected females had locked themselves out of their houses in the last three years.

Test Sam's claim at the 1% significance level.

Suppose Ari loses 31 ​% of all ping dash pong games . ​(a) What is the probability that Ari loses two ping dash pong games in a​ row? ​(b) What is the probability that Ari loses six ping dash pong games in a​ row? ​(c) When events are​ independent, their complements are independent as well. Use this result to determine the probability that Ari loses six ping dash pong games in a​ row, but does not lose seven in a row.

​(a) The probability that Ari loses two ping dash pong games in a row is . 0961 . ​(Round to four decimal places as​ needed.) ​(.31)^2

(b) The probability that Ari loses six ping dash pong games in a row is . 0009 . ​(Round to four decimal places as​ needed.) ​(.31)^6

(c) The probability that Ari loses six ping dash pong games in a​ row, but does not lose seven in a row is nothing . ​(Round to four decimal places as​ needed.)

Use the tree diagram technique that can be found in Lecture 12 to solve this problem. Suppose that 50% of all people who take a pregnancy test are, in fact, pregnant. A certain pregnancy test is known to identify 95% of pregnancies with a positive test result. However, 2% of people who are not pregnant will have a positive test result.

Find the following probabilities:

a. What is the probability that someone is pregnant and tests positive?

b. What is the overall probability of a positive test result?

c. If a woman has a positive test result, what is the probability that she is actually pregnant?

b) test statistic

c) RR

d) p-value

e) test decision

10. The population mean of annual salary for plumbers is $46,700, with a standard deviation of $5600. A random sample of 42 plumbers is drawn from this population. Find the probability that the mean salary of the sample is (a) less than $44,000. (b) between $40,000 and $51,000 (c) more than $55,000

A box contains 5 chips marked 1, 2, 3, 4, and 5. One chip is drawn at random, the number on it is noted, and the chip is replaced. The process is repeated with another chip. Let X1, X2, X3 and X4 the outcomes of the three draws which can be viewed as a random sample of size 4 from a uniform distribution on integers.

a) Calculate the Skewness and Kurtosis for the sample mean (X bar ). Explain your results

A haulage firm has 24 lorries, 15 of these are articulated

i) What is the probability that 3 randomly selected lorries are articulated ?

ii) what is the probability that no articulated lorries would be chosen if 4 lorries were randomly selected?

iii) what is the probability that more than 3 articulated lorries would be chosen if 6 were randomly selected?

Q) If holes drilled in a metal plate have a normal distribution with a mean diameter of 16.4mm and a standard deviation of 0.8mm, and a set of bolts have a normal distribution with a mean diameter of 16mm and a standard deviation of 0.5mm, what is the probability of randomly selected bolt fitting a randomly selected hole?