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A polling organization plans to ask a random sample of likely voters who they plan to vote for in an upcoming election. The researchers will report the sample proportion p̂ that favors the incumbent as an estimate of the population proportion p that favors the incumbent. What does it mean to say that p̂ is an unbiased estimator of p?
If we chose many SRSs and calculated the sample proportion p̂ for each sample, the distribution of p̂ would be centered at the value of p.
Suppose a farmer wants to estimate the true variance in daily milk yield from the dairy cows on her farm. The farmer randomly selects five dairy cows from those on her farm. She records the total amount of milk each cow yields in a single day. The farmer then computes the variance in daily milk yield from the dairy cows among these five cows.
There are nine justices currently serving the in the United States Supreme Court, and 44% of them were appointed after the year 2000. The 44% here is a
parameter
In 1936, Literary Digest polled 2.3 million adults in the United States, and 57% of them said they were vote for Alf Landon for the Presidency. The 57% here is a
statistic
The survey results from 12 local parks show the mean height of 60 ft for mature oak trees. The mean height of 60 ft is a
statistic
The 36 students in Ms. Dunham’s second grade class have a mean height of 64 inches with a standard deviation of 1.3 inches. The 1.3 inches is a
parameter
Parameter or statistic?
The proportion of business owners in the United States who attend business conferences each year
Parameter
Parameter or statistic?
The percentage of rental properties in every country whose maintenance staff live onsite
Parameter
Parameter or statistic?
The mean height of adult males throughout the world
Parameter
Parameter or statistic?
of the 100 athletic professionals in four different sports surveyed, the proportion who exercise twice a day
Statistic
Parameter or statistic?
out of 70 households that use Crest toothpaste chosen at random in Ohio, the proportion that also use mouthwash daily
Statistic
A statistic is an unbiased estimator of a parameter when
In many samples, the values of the statistic are centered at the value of the parameter
In a residential neighborhood, the median value of a house if $200,000. For which of the following sample sizes is the sample median most likely to be above $250,000?
n = 10
(Four histograms, yellow background, orange bars)
The figure shows approximate sampling distributions of 4 different statistics intended to estimate the same parameter. Which statistics are unbiased estimators? Justify your answer.
(ii) and (iii). The means of their sampling distributions appear to be equal to the corresponding population parameters.
(Four histograms, yellow background, orange bars)
Which statistic does the best job of estimating the parameter? Explain your answer.
(ii). It is unbiased and has very little variability.
(Image pictured is a large dot plot with red dots)
What would you conclude about the thermostat manufacturer’s claim? Explain your reasoning.
A sample standard deviation of 5°F provides convincing evidence that the manufacturer’s claim is false and that the thermostat actually has more variability than claimed.
On Tuesday, the bottles of Arizona Iced Tea filled in a plant were supposed to contain an average of 20 ounces of iced tea. Quality control inspectors selected 50 bottles at random from the day’s production. These bottles contained an average of 19.6 ounces of iced tea. Identify the sample.
the 50 bottles of tea selected at random from the day’s production
Tom is cooking a large turkey for a holiday meal. He wants to be sure that the turkey is safe to eat, which requires a minimum internal temperature of 165°F. Tom uses a thermometer to measure the temperature of the turkey meat at four randomly chosen points. The minimum reading is 170°F. Identify the population of interest.
All points in the turkey.
The information below is from the small population of 5 students. The 10 possible SRSs of size n = 2 are:
(all possible samples of students are listed)
Calculate the proportion of females for each sample, and display the sampling distribution for he sample proportion in a dotplot.
A candy maker offers Child and Adult bags of jelly beans with different color mixes. The company claims that the Child mix has 30% red jelly beans, while the adult mix has 15% red jelly beans. Assume that the candy maker’s claim is true. Suppose we take a random sample of 50 jelly beans from the Child mix and a separate random sample of 100 jelly beans from the Adult mix. Let…
The 10% condition is met because 50 is less than 10% of the jelly beans in the Child mix and 100 is less than 10% of the jelly beans in the Adult mix
(less than, Child mix, less than, Adult mix…)
(Same Child and Adult mix jelly bean question)
Calculate and interpret the standard deviation of the sampling distribution.
The difference (Child mix - Adult mix) in the sample proportions of red jelly beans typically varies by about 0.0740 from the true difference in proportions of 0.15.
(Same Child and Adult mix jelly bean question)
What is the shape of the sampling distribution of 𝑝̂ 𝐶−𝑝̂ 𝐴p^C−p^A? Why?
Approximately Normal because the expected number of successes and failures for each group (𝑛𝐶𝑝𝐶,𝑛𝐶(1−𝑝𝐶),𝑛𝐴𝑝𝐴,and 𝑛𝐴(1−𝑝𝐴)(nCpC,nC(1−pC),nApA,and nA(1−pA) are all at least 10.
(Same Child and Adult mix jelly bean question)
What is the mean of the sampling distribution of 𝑝̂ 𝐶−𝑝̂ 𝐴p^C−p^A?
0.15
(Same Child and Adult mix jelly bean question)
The probability that the proportion of red jelly beans in the Child sample is less than or equal to the proportion of red jelly beans in the adult sample, assuming that the company’s claim is true is 0.0212. Suppose that the Child and Adult samples contain an qual proportion of red jelly beans. Does the probability, 0.0212, give you reason to doubt the company’s claim? Explain your reasoning.
Yes. There is only a 2% chance of getting a proportion of red jelly beans in the child sample less than or equal to the proportion of red jelly beans in the adult sample if the company’s claim is true.
(Same Child and Adult mix jelly bean question)
Find the probability that the proportion of red jelly beans in the Child sample is less than or qual to the proportion of red jelly beans in the Adult sample, assuming that the company’s claim is true.
0.0213
A factory employs 3000 unionized workers, 90% of whom are male. A random sample of 15 workers is selected for a survey about worker satisfaction. Let p̂ = the proportion of males in the sample. Describe the shape of sampling distribution p̂. Justify your answer.
np = 15(0.9) = 13.5 ≥ 10, but n (1-p) = 15 (1-0.9) = 1.5 < 10, so the sampling distribution of p̂ is not approximately normal. Because p = 0.90 is closer to 1 than to 0, the sampling distribution of p̂ is skewed to the left.
The magazine Sports Illustrated asked a random sample of 740 Division I college athletes, “Do you believe performance enhancing drugs are a problem in college sports?” Suppose that 30% of all Division I athletes think that these drugs are a problem. Let p̂ by the sample proportion who say that these drugs are a problem. Which of the following are the mean and standard deviation of the sampling distribution of the sample proportion p?
Mean = 0.30, SD = 0.017
(Same Sports Illustrated question)
The sampling distribution of p̂ is approximately Normal because
np = 225 and n (1-p) = 525 are both at least 10
Your mail-order company advertises that is ships 90% of its orders within three working days. You select an SRS or 100 of the 5000 orders received in the past week for an audit. The audit reveals that 86 of these orders were shipped on time. If the company really ships 90% of its orders on time, the probability that the proportion in an SRS of 100 orders is 0.86 or less was found to be 0.0912. Is there convincing evidence that less than 90% of al orders from this company are shipped within three working days? Explain your reasoning.
No. It isn’t unusual to get a sample proportion of 0.86 or smaller when selecting an SRS of 100 from a population in which p = 0.90. It is plausible that the 90% claim is correct and that the lower than expected percentage is due to chance alone.
(Same mail-order company advertisement question)
If the company really ships 90% of its orders on time, what is the probability that the proportion in an SRS of 100 orders is 0.86 or less?
P (p̂ ≤ 0.86) = 0.0912
A machine is designed to fill 16-ounce bottles of shampoo. When the machine is working properly, the amount poured into the bottles follows a Normal distribution with mean 16.05 ounces and standard deviation 0.1 ounce. Assume that the machine is working properly.
If 4 bottles are randomly selected and the number of ounces in each bottle is measured, then, there is about a 95% probability that the sample mean will fall in which of the following intervals?
15.95 to 16.15 ounces
The distribution of scores on the mathematics part of the SAT exam in a recent year was approximately Normal with mean 515 and standard deviation 114 . Imagine choosing many SRSs of 100 students who took the exam and averaging their SAT Math scores.
Which of the following are the mean and standard deviation od the sampling distribution of x̄?
Mean = 515, SD 114/√100
The level of cholesterol in the blood for all men aged 20 to 34 to follows a Normal distribution with mean 118 milligrams per deciliter (mg/dl) and a standard deviation 41 mg/dl. 14-year-old boys, and blood cholesterol levels follow a Normal distribution with a mean mg/dl and a standard deviation of mg/dl.
