BIO240 Spring 2020
Instructions: Please type your answers in this word document in blue text. Following each question, the maximum line limits for an answer are specified in parentheses. For example, (2LN) means that the answer must be two lines or less. You may not alter fonts/font size of your responses.
As we covered in class, you may use the book, internet sources, and consultations with classmates or friends to assist you in conducting your homework. However, all work must be your own. For instance, the specific wording you use in questions must be yours. You cannot copy and paste code directly from a friend (even though your code may be quite similar). Showing your work/reasoning may allow partial credit for answers.
When completing corrections, please enter your corrections in highlighted yellow. For corrections, you may not consult with classmates. Your corrections should be placed below your original answer (please do not modify this) and must include your reasoning as to why your original answer was incorrect or why you have corrected your answer.
For all calculations you may calculate by hand or use R unless otherwise noted. If you use R you must include your code. Please show your work for calculations by hand.
5.30 True or false. Determine if the following statements are true or false, and explain your reasoning. If false, state how it could be corrected. (2 LN each)
(a) If a given value (for example, the null hypothesized value of a parameter) is within a 95% confidence interval, it will also be within a 99% confidence interval.
(b) Decreasing the significance level (α) will increase the probability of making a Type 1 Error.
(c) Suppose the null hypothesis is p = 0.5 and we fail to reject H0. Under this scenario, the true population proportion is 0.5.
(d) With large sample sizes, even small differences between the null value and the observed point estimate, difference often called the effect sizes, can be identified as statistically significant.
7.4 Find the p-value, Part II. An independent random sample is selected from an approximately normal population with an unknown standard deviation. Find the p-value for the given sample size and test statistic.
Also determine if the null hypothesis would be rejected at α = 0.01.
(a) n = 26, T = 2.485
(b) n = 18, T = 0.5
7.10 t? vs. z?. For a given confidence level, t*df is larger than z*. Explain how t*df being slightly larger than z* affects the width of the confidence interval. (5 LN)
7.12 Auto exhaust and lead exposure. Researchers interested in lead exposure due to car exhaust sampled the blood of 52 police officers subjected to constant inhalation of automobile exhaust fumes while working traffic enforcement in a primarily urban environment. The blood samples of these officers had an average lead concentration of 124.32 µg/l and an SD of 37.74 µg/l; a previous study of individuals from a nearby suburb, with no history of exposure, found an average blood level concentration of 35 µg/l.
(a) Write down the hypotheses in notation that would be appropriate for testing if the police officers appear to have been exposed to a different concentration of lead. (2 LN)
Null or Alternative? Identify whether each of the following statements is more appropriate as the null hypothesis or as the alternative hypothesis in a test:
a.) Hypothesis: The number of hours preschool children spend watching television affects how they behave with other children when at day care.
b.) Hypothesis: Most genetic mutations are deleterious to health.
c.) Hypothesis: A diet of fast foods has no effect on liver function.
d.) Hypothesis: Cigarette smoking influences risk of suicide.
e.) Hypothesis: Growth rates of forest trees are unaffected by increases in carbon dioxide levels in the atmosphere.
(a) The probability of committing a Type I error?
(b) The probability of committing a Type II error?
(c) The significance level (α)?
Species elevation shift. As the world warms, the geographic ranges of species might shift, potentially towards cooler areas. Chen et al. (2011) described recent changes in the highest elevation at which species occur. Typically, higher elevations are cooler than lower elevations. Below are the changes in highest elevation for 31 taxa, in meters, over the late 1900s and early 2000s. Many taxa were surveyed from the literature, including plants, vertebrates, and arthropods. Positive numbers indicate upward shifts in elevation, and negative numbers indicate shifts to lower elevations. Answer the following questions in R using the data below and be sure to label any plots appropriately.
Data: 58.9, 7.8, 108.6, 44.8, 11.1, 19.2, 61.9, 30.5, 12.7, 35.8, 7.4, 39.3, 24.0, 62.1, 24.3, 55.3, 32.7, 65.3, -19.3, 7.6, -5.2, -2.1, 31.0, 69.0, 88.6, 39.5, 20.7, 89.0, 69.0, 64.9, 64.8
a.) What are the mean, sample size, and standard deviation for these data?
b.) What is the standard error of the mean for these data?
c.) How many degrees of freedom will be associated with a confidence interval and a one-sample t-test for the mean elevation shift?
d.) What value of α is needed for a 95% confidence interval?
e.) What is the critical value of t (t*) for this α and number of degrees of freedom?
f.) Using R, check all assumptions, generating any required graphs, necessary to be able to calculate a confidence interval and ensure that the conditions for inference have been satisfied.
g.) Calculate the 95% confidence interval for the mean using these data.
h.) For a one sample t-test, write the appropriate null and alternative hypotheses.
i.) Conduct a t-test to test these hypotheses. Report the value of T and associated p-value for this test.
j.) Interpret the p-value—did species change their highest elevation on average?
7.28 Fuel efficiency of manual and automatic cars, Part I. Each year the US Environmental Protection Agency (EPA) releases fuel economy data on cars manufactured in that year. Below are summary statistics on fuel efficiency (in miles/gallon) from random samples of cars with manual and automatic transmissions. Do these data provide strong evidence of a difference between the average fuel efficiency of cars with manual and automatic transmissions in terms of their average city mileage? Assume that conditions for inference are satisfied. Be sure to write your hypotheses, analyze data using the most appropriate testing approach and interpret the results of your test.