# Homework assignment

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Hi, I hope all is well. Please look at the 2 files attached and do them. Let me know if you need anything. Thanks again for helping me. I will also need your help with other assignments. Thanks again.

Homework 3 (50 pts) Due 6/29 1. In an agricultural experiment, a large field of wheat was divided into many plots (each plot being 7 x 100 ft) and the yield of grain was measured for each plot. These plot yields approximately followed a normal distribution with mean 70 lbs. and SD 8 lbs. a) Let Y represent the yield rate of a randomly selected plot. Find the percentage of plots with a yield rate greater than 80 lbs. b) Let M1 represent the mean yield rate of 5 plots chosen at random from the field. According to CLT, what is the mean and standard deviation of M1? And find the probability of getting the average yield rate greater than 80 lbs. c) Let M2 represent the mean yield rate of 25 plots chosen at random from the field, According to CLT, what is the mean and standard deviation of M1? And find the probability of getting the average yield rate greater than 80 lbs. d) Does the sample size affect the use of the Central Limit Theorem in b) and c)? Why or why not? 2. The survival times of guinea pigs inoculated with an infectious viral strain vary from animal to animal. The distribution of survival times is strongly skewed to the right. The central limit theorem says that a) as we study more an more infected guinea pigs, their average survival time gets ever closer to the mean Β΅ for all guinea pigs. b) the average survival time of a large number of infected guinea pigs has a distribution of the same shape (strongly skewed) as the distribution for individual infected guinea pigs. c) the average survival time of a large number of infected guinea pigs has a distribution that is close to Normal. You answer is_________ 3. The shell of the land snail Limocolaria martensia has two possible color forms: streaked and pallid. In a certain population of these snails, 65% of the individuals have streaked shells. Suppose a random sample of 100 snails is to be chosen from the population, and let πΜ be the sample proportion of streaked snails. a). Is the sampling distribution of πΜ approximately Normal? If so, what is its mean and standard deviation? b). Find the percentage of samples for which πΜ is within 5% of the population proportion (65%), i.e., Pr(60% < πΜ < 70%). c). Suppose the sample size is reduced to 50, do you think Pr(60% < πΜ < 70%) is increased or decreased? Why? Verify your statement by computing this probability. 4. The 2010 census found that 13.9% of adults in the US identified themselves as of Hispanic origin. An opinion poll plans to interview 1500 adults at random. a) What are the mean and standard deviation of the sampling distribution of the proportion of individuals of Hispanic origin for samples of this size? b) Fin the probability that such a sample will contain 12% or fewer individuals of Hispanic origin. 5. A dendritic tree is a branched structure that emanates from the body of a nerve cell. In a study of brain development, researchers examined brain tissue from 36 adult guinea pigs. The investigators randomly selected nerve cells from a certain region of the brain and counted the number of dendritic branch segments emanating from each selected cell. A total of 36 cells (1 per pig) was selected, and the resulting counts were as follows: measurement 28 24 38 25 42 26 29 45 25 26 49 28 35 47 26 31 35 28 41 46 33 24 26 32 48 35 35 39 53 38 38 59 17 26 44 53 a) Construct a 95% confidence interval for the population mean of dendrite branch segments (The variable can be seen as continuous in this case). b) Suppose you want to estimate the population mean to be within 2 cells, how many pigs should be sampled? 6. [PQ] Use JMP to find the 80%, 90% and 95% confident interval for the population mean of dendrite branch segments in question 5.
Homework 4 (50 pts) Due 7/6 1. On average, high school juniors who retake the SAT exam as seniors improve their Mathematics score by 13 points. One professor has designed a rigorous coaching program that she thinks will produce a larger increase. A group of juniors goes through this program and retakes the SAT as seniors. Their scores improve by an average of 15 points. Assuming the students in the program are an simple random sample (SRS) of the population of all juniors who take the SAT, and the distribution of Math score gains in the entire population is Normal with unknown mean ΞΌ and a known standard deviation π =30. There are 50 students in the sample. To test H0 π = 13, Ha π > 13 a). Compute the test statistic with the z method, zs = πΜβπ0 π βπ . b) The test statistic zs follows _________distribution. A) standard normal (0, 1) B) normal (ΞΌ, Ο) C) normal (ΞΌ, Ο/βn) D) t(n-1) b). Compute critical value, zΞ± from the z table, at Ξ± = 0.05; Then decide the reject region zs > zΞ± c). Compute the pvalue. p-value= Pr( getting at least as extreme as current sample | Ho is true) =Pr(π§π  > ππ’πππππ‘ π‘ππ π‘ π π‘ππ‘ππ π‘ππ if ΞΌ = 13) 2. You are testing Ho: ΞΌ = 10 against Ha: π > 10 based on a SRS of 20 observations from a Normal population. The data give π₯Μ = 12 and s=4. a). The value of test statistic is __________________________ b). The degree of freedom for this statistic is _______________ c). Estimating from the T table, the pvalue for the statistic is __________________ d). We use the t method in this problem. Because the t method is robust, the most important condition for its safe use is that A) the population standard deviation Ο is known. B) the population distribution must be exactly Normal. C) the data can be regarded as an SRS from the population. 3. A researcher is looking at the relative abundance of microbiota in the guts of wild chimpanzees in Cameroon. He randomly selects 25 chimpanzees from the population and analyzes their fecal samples for microbiota abundance. The data is below. Chimp ID # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Relative abundance microbiota 0.08 0.12 0.23 0.11 0.13 0.22 0.08 0.11 0.15 0.18 0.09 0.10 0.20 0.20 0.20 0.12 0.19 0.17 0.20 0.18 0.05 0.09 0.14 0.22 0.16 Given alpha = 0.05, perform a hypothesis test to address each of the following questions. Remember to define Ho and Ha, compute the test statistic, define the reject region, estimate the p-value and then state the conclusion. Is the population mean of microbiota relative abundance different from 0.15? 4. A research lab is testing effectiveness of a medicine on zebrafish. The medicine is used to help the fish swim more efficiently by helping repair damage to its muscle tissue. Fish are put in a machine and gain a βpointβ for each movement they make. 10 17 18 18 21 Number of movements (or points) for a sample of 35 fish: 15 16 21 13 18 19 11 13 12 22 23 16 19 15 14 12 22 15 16 17 14 19 16 11 19 26 20 17 20 21 a) Construct a 95% CI for the population mean number of movements. b) Is the number of movements less than 16? Given alpha = 0.05, perform a hypothesis test to answer this question. Define Ho and Ha, compute the test statistic, define the reject region, estimate the pvalue, and state the conclusion. 5. [PQ] Assumption checking and data transformation to use the T-method for both Confidence Interval and Hypothesis Testing. Go through the Siblings case study, then use JMP to solve question 3 and 4 in this homework. a) Report the confidence interval from the JMP output. b) Follow the steps to conduct a T-test: i) Define the hypotheses with the correct symbols. ii) Perform a Normality check on data iii) If the data is Normal, perform the T-test. Show the (relevant) JMP output and highlight the exact p-value. iv) Interpret the p-value and state the conclusion. 6. (This is also the Sweetening cola example in topic 8): calculating power. The cola maker of Example 15.7 wants to test at the 5% significance level the following hypotheses: H0: ΞΌ = 0 versus Ha: ΞΌ > 0 Ten taste scores were used for the significance test. The distribution of taste scores is assumed to be roughly Normal with standard deviation Ο = 1. We want to calculate the power of this test when the true mean sweetness loss is ΞΌ = 0.8. (a) What values of the z statistic would lead us to reject H0? To what values of correspond? Use the inverse Normal calculations to figure this out. do they (b) When the true mean sweetness loss is ΞΌ = 0.8, how often would we reject H0? That is, what is the probability of obtaining sample averages like the ones defined in (a) when ΞΌ = 0.8? Use the z table to calculate this probability. This probability is the power of your test against the alternative ΞΌ = 0.8, the probability of rejecting H0 when the alternative ΞΌ = 0.8 is true. (c) If the sample size is increased to 20, compute the power. (d) If the significant level is decreased to 1%, compute the power.

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School: University of Maryland

Attached.

Homework 3 (50 pts) Due 6/29

1. In an agricultural experiment, a large field of wheat was divided into many plots
(each plot being 7 x 100 ft) and the yield of grain was measured for each plot. These plot yields
approximately followed a normal distribution with mean 70 lbs. and SD 8 lbs.
a) Let Y represent the yield rate of a randomly selected plot. Find the percentage of plots
with a yield rate greater than 80 lbs.
80 β π₯Μ
π(π > 80) = π (π§ >
) = π(π§ > 1.25) = 1 β 0.8944 = 0.1056
π
10.56%
b) Let M1represent the mean yield rate of 5 plots chosen at random from the
field.According to CLT, what is the mean and standard deviation of M1? And find the
probability of getting the average yield rate greater than 80 lbs.
π
π₯Μ = 70 π2 =
= 3.58
β5
80 β π₯Μ
π(π1 > 80) = π (π§ >
) = π(π§ > 2.79) = 1 β 0.9974 = 0.0026
π
c) Let M2 represent the mean yield rate of 25 plots chosen at random from the field,
According to CLT, what is the mean and standard deviation of M1? And find the
probability of getting the average yield rate greater than 80 lbs.
π₯Μ = 70 π2 =

π

= 1.6
β25
80 β π₯Μ
π(π2 > 80) = π (π§ >
) = π(π§ > 6.25) = 0
π
d) Does the sample size affect the use ofthe Central Limit Theorem in b) and c)?Why or
why not?
Yes it does, because CLT should be used if the sample size is more than 25.
2. The survival times of guinea pigs inoculated with an infectious viral strain vary from anim...

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Anonymous
Wow this is really good.... didn't expect it. Sweet!!!!

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