BIOM301 Exam 2

You want to know if the average number of rainy days in Maryland is significantly less than the historic value of '114'. You find that for the last 10 years the average is 107 with a sample standard deviation of 5. Which statistical test will you use to analyze your data?

1 population t-test for the mean (question about the mean and sample std deviation given so 1 pop ttest for mu)

Have the percent of UMD students holding a job while taking full time classes significantly changed during the pandemic? Before the pandemic the known, long-term value for UMD students was 38%. You sample 100 students and find that the current percent is 52%. Which statistical test will you use to analyze your data?

1 population z-test for percent (question about percent, so z test for percent)

Random variables require:

1. every outcome is independent is every other outcome 2. one outcome occurs with every run of the experiment 3. outcomes are randomly generated

What is the level of probability (in percent) required for some event outcome to be considered rare or unusual?

5%

You are testing a new drug to treat hemorrhagic fever. If you get the dose wrong people could die. Which of the following is probably the best level to use for your confidence interval for the correct dosage of drug?

99% (if critical, go with the highest level of confidence)

If our decision in a hypothesis test is to fail to reject the null hypothesis, then we know that the null hypothesis must be true.

False

If your test statistic falls in the 'critical region' you will accept the null hypothesis

False

The histograms of all sampling distributions of sample means will be symmetrical.

False

s the sample size increases, the standard deviation of the population sampled will decrease.

False

Increasing your sample size will decrease the value of your population standard deviation

False (changing population size doesn't affect the population in any way it only changes the SDSM)

You generate a 95% confidence interval for 𝜇μ. You are exactly 95% confident that the interval includes the sample mean

False (you are 100% sure it includes the sample mean because the sample mean is in the center of your CI, you are only 95% sure it includes the population mean which is represented by mu (μ))

A confidence interval is an example of a point estimate

False (a confidence interval is an interval estimate, point estimates are things like the sample mean)

Sampling Error and Standard Error have the exact same definitions

False (sampling error is a general term, std error = sigma divided by sqrt of n)

What population parameters do you need to know to estimate the shape and spread of a normal distribution?

Mu = population mean, and sigma or sigma squared = population standard deviation or variance

The probability of all possible outcomes for some discrete random probability experiment must equal a value of 1.0 or 100%

True

The standard error is equal to the standard deviation of the sampling distribution of sample means.

True

The standard normal curve is always symmetric around a value of 0.

True

The variablity of sample mean values is estimated by the 'standard error'

True

When sample sizes are small, the t-distribution is less peaked and more spread out than the z-distribution

True

William Gossett used empirical data to determine the original t-distribution

True

z - values are associated with the Standard Normal Curve

True

Samples taken from the same population will vary in the values of their sample means

True (due to chance and because your sample size is much smaller than the population size, the sample means will vary among samples from the same population)

You have a population that is normally distributed with a population mean weight of 30 grams and a population standard deviation of 5. For this distribution, the P(X>30) will be exactly the same as P(X>30)

True (for continuous variables, the difference between these 2 questions is the P(X=30) which is a straight line on this probability curve. A straight line has an area of 0, so it doesn't change the probability if it is added or left out)

A population is normally distributed with a mean of 200 and a standard deviation of 40. What is the probability of randomly sampling and getting a value between 210 and 266? P(210<X<266)? Give your answer as 0.XXXX

0.3518

A population is normally distributed with a mean of 500 and a standard deviation of 70. What is the probability of randomly sampling and getting a value between 500 and 600? P(500<X<600)? Give your answer as 0.XXXX

0.4236

The probability of liking dogs is 60%. For 200 people, what is the probability that 115 OR MORE people like dogs in any order? Give your answer as 0.XXXX

0.7642

A population is normally distributed with a mean of 50 and a standard deviation of 6. What is the probability of randomly sampling and getting a value between 41 and 59? P(41<X<59)? Give your answer as 0.XXXX

0.8664

The population mean of the sampling distribution of sample means will equal the population mean even if the population is not normally distributed

True (the SDSM is all possible sample means that could be taken from the population given a specific sample size. The sample means will still vary around the population mean because that is the center of the distribution)

Even if your t* value is in the t critical region it is still possible that the null hypothesis is really true in the population.

True (you can always make a mistake since you are using a sample to make inference to the population. If the sample doesn't match the population you will make some type of error. This is a Type I error)

The more variable your population is, the larger your standard error will also be

True (std error = sigma divided by the sqrt of n so the bigger sigma is the larger the std error also is)

When sample sizes are small, the t-distribution changes to require a greater level of statistical proof before you can reject the null hypothesis.

True - the distribution is less peaked and more spread out so you need a larger (positive or negative) t* value to reject the null

A population is normally distributed with a mean of 20. What is the population standard deviation if you know 18% of the values are less than a value of 15?

between 5.35 and 5.8

for #9 above, the hypothesis is:

both tailed want to know if changed, so could go up or down

Why does increasing your sample size decrease the impact of outliers on the sample mean?

a large outlier will be offset by a small outlier (large outliers offset small ones, so the sample mean is closer to the true population mean and not skewed towards the outlier)

If you reject the null hypothesis, you could have made:

a type I error (if you reject you could only have made a type I error (rejecting based on sample data when in the population the null hypothesis is not false))

for #7 above, the hypothesis is:

left tailed you want to know if the number days is significantly less, since equation would be set up as xbar - SV, you will only reject if your t* value is in the left tail

You are in charge of the online store for the Animal Sciences Department. You collect the following data about the 100 items sold in December. What was the average value of an item sold at the store during that time? Give your answer as XX.X

mu = sum of x * P(x) = (20*.5) + (30*.2) + (10*.1) + (5*.1) + (15+.1) = 19

What population parameters do you need to know to estimate the shape and spread of a binomial distribution?

n = sample size, p = probability of success (not q since if you know p you know q)

You can assume that the sampling distribution of sample means is normally distributed if:

n > 30 (it has to be greater than 30 before the assumption kicks in)

Decreasing the sample mean will cause the confidence interval to

not change

You are analyzing data using a 1 population z test for the mean. If you z* value is closer to zero than the z critical value:

you cannot reject the null hypothesis and your p-value will be > or = 0.05 (if z* is close to zero, you have no evidence to reject the null hypotheses -- but we never say accept the null, only reject or don't reject it) if you don't reject then > 0.05 (I don't know why elms won't let me underline symbols in the answers)

For the standard normal curve, what number value divides the upper 50% of the values from the rest?

the Mean

Probability assumes the population values are known and you are calculating the likelihood of possible samples that could be collected. Because of this, the standard deviation is reported as

the population standard deviation = σ (population parameter is the population standard deviation = σ)

When graphing sample means in a journal article, what is the error bar shown on these graphs representative of?

the standard error (research hopes to make inference to the population that was sampled AND to think about not just this sample but other samples that COULD have been collected By using the std error, we are reflecting that consideration)

The risk of a Type I error is directly controlled in a hypothesis test by establishing a level for α.

true

Why is the t-distribution different from the z-distribution?

when sample sizes are small, you need a greater burden of proof to reject the null hypothesis (you get the greater burden of proof by requiring a more extreme t* value when sample size is small compared to the z* value.)

You test whether dogs or cats were more likely to be adopted in the last month. You compare the mean numbers for all DC area adoption agencies. What is the correct alternative hypothesis?

µ dogs ≠ µ cats (I don't specify which is adopted more so it is a 2 tailed question and the alternative is not equals)

You want to know if a new drug leads to greater weight gain in mice (gms). The correct alternative hypothesis is:

µ new drug > µ old drug (the statement of interest (new drug better) should be in the Ha so new > old)

If a population is not normally distributed, the sampling distribution of sample means will appear normal if the sample size used is at least___

31

If the population is normally distributed then the sampling distribution of sample means will also always be normally distributed.

True

Normal distributions extend from negative infinity to positive infinity

True

Only the t and not the z distribution varies with sample size.

True

A continuous random variable could have any possible distribution shape

True

All standard normal curves are normally distributed

True

As sample size increases, a randomly selected sample will have a sample mean that is closer to the true population value

True

As sample size increases, the sampling distribution of sample means will become more peaked and less spread out.

True

Both t and z distributions are always unimodal and symmetric.

True

The probability of being left handed in some population is 20%. If 20 people are chosen at random, what is the probability that exactly 7 of the 20 are left handed in any order?

(20!/7!13!)(.2)^7(.8)^13

You have a population that is normally distributed with a population mean weight of 30 grams and a population standard deviation of 5. For this distribution what is the probability that you take a sample of size 16 and get a mean value of 32 or more? Give your answer as 0.XXXX

0.0548

A population is normally distributed with a mean of 10 and a standard deviation of 2. What is the probability of randomly sampling and getting a value between 7 and 8.5? P(7<X<8.5)? Give your answer as 0.XXXX

0.1598

You sample 40 dogs and find a sample mean weight of 60 lbs with a sample standard deviation of 5. What is the LOWER limit for a 95% confidence interval the mean weight of dogs? The statistical value you will need is 2.03. Give your answer as XX.X

= 60 - 2.03(5/sqrt of 40) = 58.4 I am 95% confident that the true population mean weight of dogs is... (need to give level of confidence about the population parameter)

For the standard normal distribution, the mean will always equal 0 and the standard deviation will be 1.

True

For a 95% confidence interval, what are you 95% confident about?

Interval includes the true population mean

The probability of randomly sampling someone in my neighborhood and finding that they own a pet is 0.7. You randomly sample 6 people in my neighborhood. What is the probability that 5 OR MORE of them own a pet in any order? Give your answer as 0.XX

P(5 out of 6) own pet = 0.3025 from previous question p = .7 q = .3 n=6, x=5 n-x = 1 6!/5!1! * .7^5 * .3^1 = .3025 ADD this to P(6 out of 6) own a pet = .7^6 = .1177 .3024 + .1177 = .4202

For the data set below, what is the P(X>1)? Give your answer as 0.XX For the data set below, what is the P(X>1)? Give your answer as 0.XX

P(X>1) = .4 + .1 = .5 P(X> or = to 1) = .2 + .4 + .1 = .7

If nothing else is influencing the distribution of a continuous variable (e.g., predation pressure, evolution, etc.) we often (but by no means always) see a distribution that is unimodal, symmetric and bell-shaped.

True

α represents the

The probability of a Type I error

Decreasing the variability in the population will cause the length of the confidence interval to

decrease

if you have lots of observations for a continuous variable, you can smooth out the frequency histogram bars into a

density curve

Decreasing the sample size will cause the length of the confidence interval to

increase

If it is absolutely critical that you don't make a type I error in your study, which of the following is under your control and can help with that issue?

increase your sample size and decrease alpha (both increasing your sample size (gives you more info about population so you can make a correct decision) and decrease alpha (so it is less likely you make a Type I error) are options under your control. You can't directly change beta)

For #3 above, you decision for level of confidence has led to a larger confidence interval for the drug dose than you would like to use. What can you do to make the interval smaller?

increase your sample size, this will lead to a smaller interval (increasing n will decrease the std error in the equation and lead to a smaller interval)

The probability of randomly sampling a student on campus and finding that they are wearing a red shirt is .3. what is the probability of sampling 10 students and finding that exactly 4 are wearing red shirts in any order? Give your answer as 0.XX

p = .3, q = 1-p = .7 n=10 x=4 n-x = 6 10!/4!6! * (.3)^4 *( .7)^6 =0.20012

The probability of randomly sampling someone in my neighborhood and finding that they own a pet is 0.7. You randomly sample 6 people in my neighborhood. What is the probability that exactly 5 of them own a pet in any order? Give your answer as 0.XX

p = .7 q = .3 n=6, x=5 n-x = 1 6!/5!1! * .7^5 * .3^1 = .3025

Now you are interested in the percent of dogs that were neutered in your sample. For 40 dogs, 30 of them were neutered. What is the LOWER limit for this confidence interval? The statistical value you will need is 1.96. Give your answer as 0.XX

p' = 30 neutered/40 dogs .75. The std error for a percent is the square root of (p'q'/n) = sqrt of ((.75*.25)/40) = .07 the lower limit would be .75 - (1.96*.07) = .75 - .13 = .62

Showing outcomes of all possible discrete probability events as a table listing the outcomes and their probabilities is:

probability Distribution (a table is a distribution. graph is histogram, mathematical relationship as an equation is function)

P(1/4) for x = 1, 2, 3, 4 is an example of a probability

probability function