(Please use Excel or SPSS to answer the following questions, if you use Excel, please copy and paste the function you used to obtain the answer)
- Sheila’s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation in both the actual glucose level and blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter (mg/dL) one hour after a sugary drink is ingested. Assume that Sheila’s measured glucose level one hour after ingesting the sugary drink varies according to the normal distribution with mg/dL and mg/dL.
- If a single glucose measurement is taken, what is the probability that Sheila will be diagnosed as having gestational diabetes?
- If measurements are taken instead on three separate occasions and the mean result of the three measurements is compared with the criterion 140 mg/dL to determine whether Sheila has gestational diabetes, what is the probability that Sheila is diagnosed as having gestational diabetes?
- Suppose that college faculty with the rank of professor at two-year institutions earn an average of $64,571 per year with a standard deviation of $4,000. In an attempt to verify this salary level, a random sample of 60 professors was selected from a personnel database for all two-year institutions in the United States and the average salary was calculated.
- Describe the sampling distribution of the sample mean
- Within what limits would you expect the sample average to lie, with probability 0.95?
- Calculate the probability that the sample mean is greater than $66,000.
- If your random sample actually produced a sample mean of $66,000, would you consider this unusual? What conclusion might you draw? Hint: what is the probability of observing a sample mean larger than this?
- A manufacturer of compact fluorescent light bulbs advertises that the distribution of the lifespans of these light bulbs is nearly normal with a mean of 9,000 hours and a standard deviation of 1,000 hours.
- What is the probability that a randomly chosen light bulb lasts more than 10,500 hours?
- Describe the distribution of the mean lifespan of 15 light bulbs. (Hint: Is it normal? What is the mean and standard deviation?)
- What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than 10,500 hours?
- Could you estimate the probabilities from parts (1) and (3) if the lifespans of light bulbs had a skewed distribution?
- Suppose an iPod has 3,000 songs, and we also know that the mean length of songs on this iPod is 3.45 minutes with standard deviation 1.63 minutes.
- What is the probability that a randomly selected song lasts more than 5 minutes?
- You are about to go for an hour run and you make a random playlist of 15 songs. What is the probability that your playlist lasts for the entire duration? (Hint: If you want the playlist to last 60 minutes, what should be the minimum average length of a song?)
- You are about to take a trip to visit your grandparents and the drive is 6 hours. You make a random playlist of 100 songs. What is the probability that your playlist lasts the entire drive?
Please use software to answer the questions below.
- Conventional wisdom has suggested that women are more talkative than men – but does this hold up in the statistical sense? One study designed to examine this stereotype collected data on the speech habits of 42 women and 37 men in the United States. Note that the gender variable indicates the gender (1=men, 2=women) and wordsper variable indicates the average number of words spoken per day for each subject. For the following questions, please use the provided information to answer the questions.
- The mean number of words spoken per day by the women was 14,297 with a standard deviation of 6441. Use the 68-95-99.7 rule to describe the distribution of the sample mean number of words spoken per day by women. (Find values between which 68% of the distribution falls, 95% like we’ve done in class)
- Do you think that applying the 68-95-99.7 rule in this situation is reasonable? Explain your answer. (For what distribution does the Empirical Rule apply?)
- The men averaged 14,060 words per day with a standard deviation of 9065. Answer the questions in parts (1) and (2) for the men.
- Do you think that the data support conventional wisdom? Explain your answer, using the figure below for comparing the two distributions. We will learn statistical inference methods later in the semester to address these type of questions formally.