Exam 3: Randomness in Data

Give an intuitive explanation for "In general, individual observations vary more than averages".

Suppose that a family has five children, consisting of three girls and two boys. Call the girls Abigail, Brianna, and Courtney; call the boys Duane and Evan. Suppose further that two of these children are to be selected at random. -Determine the expected value of the number of girls who will be selected.

Suppose you flip a fair coin until it lands heads up for the first time. It can be shown (do not try to calculate this) that the expected value of the number of flips required is 2. Explain (with a sentence or two) what this expected value means in this context.

Suppose that a phone company reports that the average duration of a cell phone call is 1.7 minutes, with a standard deviation of 1.4 minutes. -What does the CLT say about the sampling distribution of the sample mean call duration in a random sample of 60 calls?

Suppose that a family has five children, consisting of three girls and two boys. Call the girls Abigail, Brianna, and Courtney; call the boys Duane and Evan. Suppose further that two of these children are to be selected at random. -Write a sentence or two interpreting what this expected value means in this context

Suppose the number of French fries in a small bag at a particular fast-food restaurant follows a normal model with mean 49 fries and standard deviation 5 fries. -The restaurant could advertise that $99 \%$ of its bags contain at least how many fries?

Suppose that a phone company reports that the average duration of a cell phone call is 1.7 minutes, with a standard deviation of 1.4 minutes. -Would it be reasonable to use a normal distribution to model the duration of cell phone calls? Explain, based primarily on the values reported above.

Suppose that a family has five children, consisting of three girls and two boys. Call the girls Abigail, Brianna, and Courtney; call the boys Duane and Evan. Suppose further that two of these children are to be selected at random. -If both boys are selected, would it be reasonable for the girls to cry foul and claim that the selection process must not have been random? Explain your answer, and refer to a relevant probability calculation.

Suppose I tell you that I flipped a coin multiple times and got $75 \%$ heads. Would you be reasonably convinced that this was not a fair coin (where "fair" means that the coin has a 50/50 chance of landing heads or tails)? If so, explain why. If not, describe what additional information you would ask for and explain why that information is necessary.

Give brief (one-sentence) explanations for your answers to the following. -In which case are you more likely to get all heads from flipping a fair coin: if you flip a fair coin twice or if you flip a fair coin ten times?

Suppose a phone company reports that the average duration of a cell phone call is 1.7 minutes with a standard deviation of 1.4 minutes. -If you examine a random sample of 60 of your cell phone calls and find that the sample mean duration is 2.0 minutes, would it be reasonable to conclude that the average duration in the population of all your cell phone calls is greater than 1.7 minutes? Explain, based on your calculation.

Suppose that a phone company reports that the average duration of a cell phone call is 1.7 minutes, with a standard deviation of 1.4 minutes. -Draw a well-labeled sketch to "The CLT says the sampling distribution of the sample mean call duration will be approximately normal, with mean 1.7 minutes and standard deviation $1.4\sqrt{60} = .181$ ".

Suppose a phone company reports that the average duration of a cell phone call is 1.7 minutes with a standard deviation of 1.4 minutes. -Would "the sampling distribution of the average duration call will approximately follow a normal distribution with mean 1.7 minutes and standard deviation $1.4 / \sqrt{60}$ or 0.181 minutes. and Note: The sample size is greater than 30 , so you are willing to assume the shape will be approximately normal, even though the population distribution is most plausibly skewed to the right. This is appropriate as long as the skewness is not too severe. " be larger, smaller, or the same if the (population) average duration were 1.9 minutes rather than 1.7 minutes? Explain briefly.

According to the "quick facts" listed on Cal Poly's website in fall 2011, the proportion of Cal Poly undergraduates who are in the College of Business is. Suppose that you select a simple random sample of 150 Cal Poly undergraduates, and let $\hat{p}$ represent the sample proportion who are in the College of Business. -With a larger sample size, would the probability in #4 be larger, smaller, or the same? Explain your answer.

Suppose that the drying time for a certain type of paint under specified test conditions is known to be normally distributed with mean 75 minutes and standard deviation 5 minutes. Suppose also that chemists have devised a new additive that they hope will reduce the mean drying time (without changing the standard deviation). A test is then conducted to measure the drying time for a test specimen, and the company executives decide that they will be convinced that the additive is effective only if the drying time on this specimen is less than 70 minutes. a. If the additive actually has no effect at all on the drying time, what is the probability the company executives will mistakenly conclude that it is effective? Include a sketch with your calculation. b. If you want to alter the cutoff value from 70 in order to reduce the error probability in part a to .05 , what cutoff value should you choose? c. Now suppose that the standard deviation of the drying times to 65 minutes is 2 rather than 5 minutes. Without doing any new calculations, describe how this change would affect your answers to parts a and $b$ . Give an intuitive explanation for your reasoning in both cases.

Give brief (one-sentence) explanations for your answers to the following. -If you flip a fair coin (i.e., a coin with .5 probability of landing heads) twice, are you guaranteed to get one head and one tail?

Suppose $80 \%$ of the incoming email messages for a college's computer system are spam. -After implementing a new spam blocker, if it turns out that a random sample of 200 messages contains $75 \%$ spam, would this constitute fairly convincing evidence that the (population) percentage of spam has been reduced from $80 \%$ ? Explain, based on your calculation in question "Use the CLT to approximate the probability that in a random sample of 200 incoming email messages at this college, the sample proportion of these messages that are spam would exceed .75.".

Suppose a phone company reports that the average duration of a cell phone call is 1.7 minutes with a standard deviation of 1.4 minutes. -Would "the sampling distribution of the average duration call will approximately follow a normal distribution with mean 1.7 minutes and standard deviation $1.4 / \sqrt{60}$ or 0.181 minutes. and Note: The sample size is greater than 30 , so you are willing to assume the shape will be approximately normal, even though the population distribution is most plausibly skewed to the right. This is appropriate as long as the skewness is not too severe. " be larger, smaller, or the same if the (population) standard deviation of call durations were 1.8 minutes rather than 1.4 minutes? Explain briefly.

Suppose $80 \%$ of the incoming email messages for a college's computer system are spam. -Use the CLT to approximate the probability that in a random sample of 200 incoming email messages at this college, the sample proportion of these messages that are spam would exceed .75.

Suppose a phone company reports that the average duration of a cell phone call is 1.7 minutes with a standard deviation of 1.4 minutes. -Would "the sampling distribution of the average duration call will approximately follow a normal distribution with mean 1.7 minutes and standard deviation $1.4 / \sqrt{60}$ or 0.181 minutes. and Note: The sample size is greater than 30 , so you are willing to assume the shape will be approximately normal, even though the population distribution is most plausibly skewed to the right. This is appropriate as long as the skewness is not too severe. " be larger, smaller, or the same if the sample size were 40 calls rather than 60 calls? Explain briefly.