Statistics Chapter 7 Review
The Gallup Poll once asked a random sample of 1540 adults, "Do you happen to jog?" Suppose that the true proportion of all adults who job is p=0.15.
-Mean of the sampling distribution: 0.15 -SD of the sampling distribution: 0.009 -Approximately normal?: Yes, because np and n(1-p) are both greater than 10. -Normalcdf(.13, .17, 0.15, 0.009): 0.974; there is a 97.4% chance that in a random sample of 1540 adults, 13-17% are joggers.
Sale of eggs that are contaminated with salmonella can cause food poisoning in consumers. A large egg producer takes an SRS of 200 eggs from all the eggs shipped in one day. The laboratory reports that 9 of these eggs had salmonella contamination. Unknown to the producer, 3% of all eggs shipped had salmonella.
-Population: all eggs shipped in one day -Parameter: proportion of all eggs that have -salmonella (0.03) -Sample: 200 randomly selected eggs shipped that day -Statistic: 9/200=0.045=p (proportion of sample that had salmonella)
A random sample of 1000 people who signed a card saying they intended to quit smoking were contacted 9 months later. It turned out that 210 (21%) of the sampled individuals had not smoked over the past 6 months.
-Population: people who signed a card intending to quit smoking -Parameter: proportion of those people who actually quit smoking -Sample: 1000 randomly selected card signers -Statistic: proportion of the sample that actually quit smoking; 210/100, p=0.21
R7.5 Bag Check
-normalcdf(-10^10, 0.20, 0.30, 0.0458): 0.0145; 1.45% chance than the proportion of travelers that get a red light is as small or smaller than the sample's result
A machine is designed to fill 16oz bottles of shampoo. When the machine is working properly, the amount poured into the bottles follows a normal distribution with mean 16.05oz and a standard deviation 0.1oz. Assume that the machine is working properly. If four bottles are randomly selected and the number of oz in each bottle is measures, then there is about a 95% chance that the sample mean will fall in which of the following intervals?
15.95 to 16.15 oz
According to the government data, 22% of American children under the age of six live in households with incomes less than the official poverty level. A study of learning in early childhood chooses an SRS of 300 children. Find the probability that more than 20% of the sample are from poverty-level households. Be sure to check that you can use the Normal approximation.
300<10% of all children under 6 years old; np and n(1-p) are both more than 10; Mean=0.22; SD: 0.0239; normalcdf(0.2, 10^10, 0.22, 0.0239)=0.7986
The amount that households pay service providers for access to the Internet varies quite a bit, but the mean monthly fee is $38 and the SD is $10. The distribution is not Normal: many households pay a base rate for low-speed access, but some pay much more for faster connections. A sample survey asks an SRS of 500 households with Internet access how much they pay. Let x be the mean amount paid.
A. Why you can't determine the probability that the amount a randomly selected household pays for access to the Internet exceeds $39: We don't know the shape of the population distribution. B. What are the mean and SD of the sampling distribution of x? Mean: $38; SD: 0.447 C. Approximately normal, because 500>30. D. normalcdf(39, 10^10, 38, 0.447)=0.0126; 1.26% chance of being over $39
Is the sample range an unbiased estimator of the population range?
No, because the sample range is always less than the population range. If it were unbiased, the sample would be centered at the population range.
On a flight from New York to Denver, 8% of the 125 passengers were selected for random security screening before boarding. According to the TSA, 10% of passengers at this airport are chosen for random screenings.
Parameter: 10% Statistic: 8%
On Tuesday, the bottles of Arizona Iced Tea filled in a plant were supposed to contain an average of 20 oz of iced tea. Quality control inspectors sampled 50 bottles at random from the day's production. These bottles contained an average of 19.6 oz of iced tea.
Parameter: 20 oz Statistic: 19.6 oz
The Gallup Poll asked a random sample of 515 U.S. adults whether or not they believe in ghosts. Of the respondents, 160 said "Yes."
Population: All U.S. adults Parameter: proportion of all U.S. adults that believe in ghosts Sample: 515 randomly selected U.S. adults Statistic: 160/515 believe in ghosts, p=0.31
During the winter months, the temperatures outside the Starneses' cabin in Colorado can stay well below freezing (32F or 0C) for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at 50F. She wants to know how low the temperature actually gets in the cabin. A digital thermometer records the indoor temperature at 20 randomly chosen times during a given day. The minimum reading is 38F.
Population: All times during the day Parameter: Minimum temperature of times that day Sample: 20 randomly chosen times that day Statistic: Sample's minimum temperature is 38F
Explain how we could decrease the variability of the sampling distribution of the sample range.
Take larger samples.
Which of the following statements about the sampling distribution of the sample mean is incorrect? A. The standard deviation of the sampling distribution will decrease as the sample size increases. B. The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples. C. The sample mean is an unbiased estimator of the population mean. D. The sampling distribution shows how the sample mean will vary in repeated samples. E. The sampling distribution shows how the sample was distributed around the sample mean.
The sampling distribution shows how the sample was distributed around the sample mean.
Sampling distribution
of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population
statistic
a number that describes some characteristic of a sample
parameter
a number that describes some characteristic of the population
Variability of a statistic
described by the spread of its sampling distribution; this spread is determined by the sampling design and the size of the sample. Large samples give smaller spread.
Suppose that you are a student aide in the library and agree to be paid according to the "random pay" system. Each week, the librarian flips a coin. If the coin comes up heads, your pay for the week is $80. If it comes up tails, your pay for the week is $40. You work for the library for 100 weeks. Suppose we choose an SRS of 2 weeks, and calculate your average earning x. The shape of the sampling distribution of x will be
symmetric but not Normal
