WGU Academy Statistics Module 10
Suppose the mean hours slept (per day) among a sample of college students was 7.1, and a 95% confidence interval for the population mean hours slept was constructed. Which one of these could it be? (7.1, 8.1) (6.9, 7.3) (7.0, 7.3) (6.9, 7.1)
(6.9, 7.3) -The confidence interval must be centered at the sample mean (7.1 in our case), and this is the only interval of the three that satisfies this requirement.
What sample size of U.S. adults do you need, if you would like to estimate the proportion of U.S. adults who are "pro-choice" with a 2.5% margin of error (at the 95% level)?
1600 -Indeed, the sample size needed for a margin of error of 2.5% (at the 95% level) is 1 / (.025)2 = 1,600.
In a recent study 1,115 males 25 to 35 years of age were randomly chosen and asked about their exercise habits. Based on the study results, the researchers estimated that the mean time that a male 25 to 35 years of age spends exercising in a week is about 3.5 hours.
Point estimation -here we are using the data to estimate the mean time (our parameter) using a single number—3.5 hours.
A large state university conducted a study in order to estimate μ, the mean cost of textbooks per semester of a student in the university. A stratified random sample of 530 students was chosen (ensuring representation from the different majors and different classes) and it was found that the total amount spent by these students on textbooks was $225,250. Based on the results of this study, what is the point estimate for μ? -$225,250 -$425 -$530 -There is not enough information since μ is not given.
$425 -the point estimate is the sample mean (average), which is $225,250/530=$425.
Suppose the mean hours slept (per day) among a sample of first year law students was 6.1 hours. The margin of error for the 95% confidence interval for the population mean of hours slept was 1.5 hours. Which of the following is the confidence interval? (4.6, 7.6) (3.1, 9.1) (4.5, 7.5) (3.0, 9.0)
(4.6, 7.6) -The confidence interval is the sample mean ± margin of error. In this case: 6.1 ± 1.5 = (4.6, 7.6).
In order to estimate μ, the mean number hours that males in the age group 15-17 spend exercising in a week. A random sample of 256 males in the age group 15-17 was chosen and it was found that the sample mean number of hours spending exercising per week is x=4.1. Based on similar research on other age groups, it is assumed that the population standard deviation is σ=2.4. Which of the following is the 95% confidence interval for μ, the mean number hours that males in the age group 15-17 spend exercising in a week?
(3.8, 4.4) -x±2⋅σ/√n = 4.1±2⋅2.4/√256 = 4.2±0.3 = (3.8 ,4.4)
Suppose that based on a random sample, a 95% confidence interval for the mean hours slept (per day) among graduate students was found to be (6.5, 6.9). What is the margin of error of this confidence interval? 6.7 .4 .2
.2 -Indeed, since the width of the confidence interval is .4 (6.9 - 6.5), the margin of error must be .4/2 = .2
What is the margin of error of the 99% confidence interval that you just calculated? 0.02 0.04 0.08 0.45
0.04 -The structure of the confidence interval for the population proportion is p̂ ± margin of error. In this case the confidence interval is 0.45 +/- 0.04.
A Gallup poll that was conducted in July 2014 found that out of 1,013 randomly chosen U.S. adults, 456 said that they use organic food in preparing meals. Which of the following is the correct 95% confidence interval for p̂, the proportion of all U.S. adults who use organic foods in preparing meals? (Answers are rounded as needed)
0.45±0.03 - p̂=4561013=0.45 and therefore the 95% confidence interval for p̂ is:
How many smoking pregnant women should the researcher have sampled in order to reduce the margin of error to 1.5 with a 90% confidence level? 18 307.9 17 17.55 308 756
308 -n=(z*⋅σ/m)2 =(1.645*⋅16/1.5)2 = 307.9 → n = 308
What sample size of the U.S. adults do you need, if you would like to estimate the proportion of U.S. adults who are "pro-choice" with a 1.8% margin of error (at the 95% confidence level)?
3087 - 1/m^(2) = 1/0.0182 = 3,086.42→n=3,0871
If you were to use a random sample of size n = 640 U.S. adults (instead of what you found in question 1), what would the margin of error roughly be?
4% - using the conservative approach, the margin of error would roughly be 1 / sqrt(640) = .0395, which equals approximately .04
Recall the study from the previous activity on pregnant women who smoke during pregnancy. The length of human pregnancy has a standard deviation of 16 days. How many smoking, pregnant women should the researcher have sampled in order to reduce the margin of error to 1.5 while maintaining a 95% confidence level? 21.3 22 456 455 21
456 -n=(σ⋅z/m)2 = (16⋅2/1.5)2 = 455.111 → n = 456
Which of the following best describes the goal of statistical inference?
Drawing conclusions about the population based on the observed data in the sample
When the variable of interest is quantitative, the population parameter that we infer about is the population mean (μ) associated with that variable.
For example, if we are interested in studying the annual salaries in the population of teachers in a certain state, we'll choose a sample from that population. We will use the collected salary data to make an inference about μ, the mean annual salary of all teachers in that state.
When the variable of interest is categorical, the population parameter that we will infer about is the population proportion (p) associated with that variable.
For example, we want are interested in to study opinions about the death penalty among U.S. adults. Thus our variable of interest is "death penalty (in favor/against)." We choose a sample of U.S. adults and then use the collected data to make an inference about p - the proportion of U.S. adults who support the death penalty.
A company that provides coaching for the SAT claims in one of its ads: "90% of our students improve their SAT scores after attending our course." To check the company's claim, 500 students who took the company's course were sampled and it was found that 407 of them (81.4%) actually improved their SAT scores after attending the course. Based on the sample results, we have some serious doubts regarding the accuracy of the company's claim.
Hypothesis testing -in this case, we have a claim, and we are using the data collected from a sample to assess its accuracy.
Which of the following is an example of statistical inference?
In a roadside survey a random sample of 690 drivers were stopped on a weekend night. For each driver, their gender and whether or not they were driving drunk was recorded. Based on the collected data, the researchers conclude that there is no evidence that there is a relationship between gender and drunk driving in the general population of drivers.
Hypothesis Testing
In hypothesis testing, we have some claim about the population, and we checkwhether or not the data obtained from the sample provide evidence against this claim. Example 1: It was claimed that among all U.S. adults, about half are in favor of stricter gun control and about half are against it. In a recent poll of a random sample of 1,200 U.S. adults, 60% were in favor of stricter gun control. This data, therefore, provides some evidence against the claim. Example 2: It is claimed that among drivers 18-23 years of age (our population) there is no relationship between drunk driving and gender. A roadside survey collected data from a random sample of 5,000 drivers and recorded their gender and whether they were drunk. The collected data showed roughly the same percent of drunk drivers among males and among females. These data, therefore, do not give us any reason to reject the claim that drunk driving is not related to gender.
Interval Estimation
In interval estimation, we estimate an unknown parameter using an interval of values that is likely to contain the true value of that parameter (and state how confident we are that this interval indeed captures the true value of the parameter). In other words, in interval estimation we estimate an unknown parameter by an interval of plausible values for the known parameter. Example: Based on sample results, we are 95% confident that p, the proportion of U.S. adults who are in favor of stricter gun control, is between 0.57 and 0.63. In other words, we are 95% confident that p, the proportion of U.S. adults who are in favor of stricter gun control, is covered by the interval (0.57, 0.63).
Point Estimation
In point estimation, we estimate an unknown parameter using a single number that is calculated from the sample data. Example: Based on sample results, we estimate that p, the proportion of all U.S. adults who are in favor of stricter gun control, is 0.6.
Select which form of statistical inference this conclusion represents. Based on a recent poll of 1,015 U.S. households that have an internet connection, the researchers claimed with 95% confidence that among all U.S. households who have an internet connection, between 63% and 67% have a high-speed link.
Interval estimation -the proportion of households with an internet connection that have a high-speed link is estimated to be (with 95% confidence) somewhere in the interval between 63% and 67%.
Researchers from Consumer Reports chose a random sample of 100 light bulbs from a certain brand and type and tested them in their lab. Based on the results of this study the researchers estimated that the mean lifetime of all light bulbs of this type is 743 hours. More specifically, the researchers were 99% confident that the mean lifetime of light bulbs of this type is between 740 and 746 hour. Since on their packaging, the light bulbs are claimed to have mean lifetime of 750 hours, based on the data, the researchers accuse the company of false advertising. This is not a form; we suggest that you use the browse mode and read all parts of the question carefully. Which one of the following parts of the scenario represents hypothesis testing?
Since on their packaging, the light bulbs are claimed to have mean lifetime of 750 hours, based on the data, the researchers accuse the company of false advertising. Indeed, the researchers determine that the data provide evidence against the claim that mean lifetime of this type of light bulbs is 750 hours.
Below are four different situations in which a confidence interval for μ is called for. Situation A: In order to estimate μ, the mean annual salary of high-school teachers in a certain state, a random sample of 150 teachers was chosen and their average salary was found to be $38,450. From past experience, it is known that teachers' salaries have a standard deviation of $5,000. Situation B: A medical researcher wanted to estimate μ, the mean recovery time from open-heart surgery for males between the ages of 50 and 60. The researcher followed the next 15 male patients in this age group who underwent open-heart surgery in his medical institute through their recovery period. (Comment: Even though the sample was not strictly random, there is no reason to believe that the sample of "the next 15 patients" introduces any bias, so it is as good as a random sample). The mean recovery time of the 15 patients was 26 days. From the large body of research that was done in this area, it is assumed that recovery times from open-heart surgery have a standard deviation of 3 days. Situation C: In order to estimate μ, the mean score on the quantitative reasoning part of the GRE (Graduate Record Examination
Situation B -Even though the sample was selected in a manner that is not strictly random, no bias is expected. However, the sample size is small enough (less than 30) to require that the recovery times are normally distributed. Since we do not know this, we cannot use this confidence interval.
Below are four different situations where a confidence interval formula would be useful: Situation A: A marketing executive wants to estimate the average time, in days, that a watch battery will last. She tests 50 randomly selected batteries and finds that the distribution is skewed to the left, since a couple of the batteries were defective. It is known from past experience that the standard deviation is 25 days. Situation B: A college professor desires an estimate of the mean number of hours per week that full-time college students are employed. He randomly selected 250 college students and found that they worked a mean time of 18.6 hours per week. He uses previously known data for his standard deviation. Situation C: A medical researcher at a sports medicine clinic uses 35 volunteers from the clinic to study the average number of hours the typical American exercises per week. It is known that hours of exercise are normally distributed and past data give him a standard deviation of 1.2 hours. Situation D: A high-end auto manufacturer tests 5 randomly selected cars to find out the damage caused by a 5 mph crash. It is known that this distribution is normal. Assume that the standard devi
Situation C -The first requirement of a random sample was not achieved. Not only did the researcher use volunteers, but it is quite possible that patients at a sports clinic exercise more than the typical person. Without a random sample, it does not matter what the sample size or the type of distribution are.
A psychology researcher was conducting a study about newlywed heterosexual couples during the first two years of their marriage. 513 newlywed couples were randomly chosen for the study. One of the questions that the researcher was interested in was "During a typical week, how many times do you have sex?" The 513 responses had an average of 2.35 and standard deviation of 1.2. Another question that was asked is "During a typical week, how many evenings do you go out?" 171 of the couples answered that they go out more than twice a week. 2.35 is the point estimate for which of the following? -The proportion of all newlyweds who have sex more than twice in a typical week -The mean number of times all newlyweds go out during a typical week -The mean number of times all newlyweds have sex during a typical week
The mean number of times all newlyweds have sex during a typical week -we are told that the mean number of times that the newlyweds in our sample have sex in a typical week is 2.35. Therefore, 2.35 is the point estimate for the mean in the entire population (all newlyweds).
Researchers from Consumer Reports chose a random sample of 100 light bulbs from a certain brand and type and tested them in their lab. Based on the results of this study the researchers estimated that the mean lifetime of all light bulbs of this type is 743 hours. More specifically, the researchers were 99% confident that the mean lifetime of light bulbs of this type is between 740 and 746 hour. Since on their packaging, the light bulbs are claimed to have mean lifetime of 750 hours, based on the data, the researchers accuse the company of false advertising. Which one of the following parts of the scenario represents point estimation?
The researchers estimate that the mean lifetime of all light bulbs of this type is 743 hours. Indeed the researchers estimate the mean lifetime of all bulbs of this type (the unknown parameter in this case) by a single number.
Researchers from Consumer Reports chose a random sample of 100 light bulbs from a certain brand and type and tested them in their lab. Based on the results of this study the researchers estimated that the mean lifetime of all light bulbs of this type is 743 hours. More specifically, the researchers were 99% confident that the mean lifetime of light bulbs of this type is between 740 and 746 hour. Since on their packaging, the light bulbs are claimed to have mean lifetime of 750 hours, based on the data, the researchers accuse the company of false advertising. Which one of the following parts of the scenario represents interval estimation?
The researchers were 99% confident that the mean lifetime of light bulbs of this type is between 740 and 746 hours. This is indeed an example of interval estimation since the researchers estimate the mean lifetime of all light bulbs of this type (the unknown parameter in this case) by an interval of plausible values.
What is true regarding an unbiased estimator?
The sampling distribution of the unbiased estimator is centered at the parameter that it estimates. -an estimator is called unbiased if its sampling distribution is centered at the parameter it estimates. In other words, In repeated samples, an unbiased estimator sometimes overestimates and sometimes underestimates the parameter, but on average, is equal to the parameter it estimates.
Which of the following is the correct interpretation of the 95% confidence interval you found above?
We are 95% confident that the mean number of hours that males in the age group 15-17 spend exercising per week is covered by the interval (3.8, 4.4) hours. -The confidence interval is a statement about the plausible values for the mean number of hours that males in the age group 15-17 spend exercising per week.
Which of the following is the correct interpretation of the margin of error of the 95% confidence interval (0.03)? -We are 95% confident that the proportion of all U.S. adults who use organic foods is more than 0.03 away from the observed sample proportion of 0.45. -We are 95% confident that the proportion of all U.S. adults who use organic foods is no more than 0.03 away from the observed sample proportion of 0.45. -We are 95% confident that the proportion of all U.S. adults who use organic foods is exactly 0.03 away from the observed sample proportion of 0.45.
We are 95% confident that the proportion of all U.S. adults who use organic foods is no more than 0.03 away from the observed sample proportion of 0.45. -The interpretation of the margin of error (m) is that with a certain level of confidence (95% in this case) we can say that the population proportion p̂ is within m of (or, no more than m away from) the observed sample proportion p̂.
Based on survey results, the proportion of U.S. adults who use the Internet on a daily basis is .37. This point estimate would be unbiased and most accurate if the survey was based on: a random sample of 1,000 U.S. adults. a random sample of 2,500 U.S. adults. a random sample of 1,000 college students. a random sample of 2,500 college students.
a random sample of 2,500 U.S. adults. -The estimate is based on a random sample (and is therefore unbiased) and is also based on a larger sample, which makes it more accurate.
