A drastic drop of 500 points; after, a poll said 92% said they will be better in a year. What is 92%
statistic
distribution of values taken by a statistic in all possible samples of the same size from the same population
the sampling distribution of the statistic
if a statistic is used to estimate a parameter; the mean of its sampling distribution is equal to the true value of the parameter being estimated, the statistic is
unbiased
you take a random sample of size 25 from a population with mean of 120 and standard deviation of 15. Your sample has a mean of 115 and a standard deviation of 13.8. Which has a mean of 120 and a standard deviation of 3?
-the sampling distribution of the same mean
1) list what you are given
-parameter: N=25, M=120, SD = 15
-Sample: M=115, SD = 13.8
-Statistic: M=120, SD = 3
2) Finding Sample Distribution: Parameter SD/sqroot(n) = 15/sqroot25 = 15/5 = 3
To calculate standard deviation of sample distribution (SD/sqroot(N)), which conditions must be met?
-The sample size is less than 10% of population size (N)(.10)
-does not matter what shape is
the central limit therorem
regardless of shape of the populations distribution, the sampling distribution of the sample mean from sufficiently large samples will be approximately Normally distributed
A random sample of 1000 Americans said 62% were satisified with service provided by car dealer. A simple random sample of 1000 Canadians said 59% were. The sampling variability associated with these statistics is
not exactly the same, but very close
Random sample of 16 students at school how many minutes they spend on English homework; u=45 minutes, SD = 15 minutes. Which of the following best describes what we know about the sampling distribution sample?
1) List what we know: n=16, u = 45, SD = 15
2) We can infer that
-Ux=45
-Ox= 15/sq(16) = 3.75
-Dont know shape
Olives are normally distributed with a mean of 7.7g and a SD of .2g. Represent the probability that the mean weight of a random sample of 3 olives from this population is greater than 8gm?
Z> 8-7.7 / (.2/sq(3))
-put sample size under standard deviation
College freshman study mean of 7.2 hr and SD of 5.3
1) Can you calculate probability that a randomly chosen freshman studies more than 9 hours?
no, we do not know the shape (shape = N)
College freshman study mean of 7.2 hr and SD of 5.3
What is the shape of the sampling distribution of mean x for samples of 55 randomly selected freshman?
N= 55
55 is greater than 30
sample is approx normal
College freshman study mean of 7.2 hr and SD of 5.3
What are the mean and standard deviation for the mean number of hours x spent studying by an SRS of 55 freshman?
Mx = 7.2 (same as sample)
Ox = SD/sq(N) = 5.3/sq(55) = .715
College freshman study mean of 7.2 hr and SD of 5.3
Find the probability that the average number of hours spent studying by an SRS of 55 students is greater than 9 hours
Ox= SD/sq(N) = 5.3/sq(55) = .715
Z > (9 - 7.2) / .715
In a game, a heads is to his advantage. You test a coin for fairness; 50 times and if probability is .50 it is fair
-Assume it is .50. If p^ is the proportion of heads in 50 flips of the coin, what are the mean and SD of the sampling distribution of ^p?
Up = P=.05
Op = sq ( (.5)(.5)/ 50 ) = .071
In a game, a heads is to his advantage. You test a coin for fairness; 50 times and if probability is .50 it is fair
-Explain why you can use the formula for the standard deviation of ^p in this setting
Possible coin flips is infinite; dont need to worry about 10% condition
In a game, a heads is to his advantage. You test a coin for fairness; 50 times and if probability is .50 it is fair
-You flip the coin 50 times and get 30 heads. Will you insult your friend?
30/50 = .6 = p^
P(P^ > .6) = Z > .6 - .5 / .071
P (Z > 141)
= .793
8% of the time we get this many more heads
19% of cars made in 2007 were white. In a random sample of 100 cars parked in an airport, 22% of cars were right. Identify parameter and statistic.
-19% is parameter
-22% is statistic
Best statistic for estimating a parameter
-low bias, low variability
a probability distribution that describes the relative likelihood of all possible values of a statistic
a sample distribution
In a large population, mean IQ is 112 with SD 20. 200 adults are randomly selected for a market research campaign. Sampling distribution of sample mean IQ is
-approx normal
-mean 112 (given)
-SD 1.414
If we take two samples from two different schools, what should the relative size of our samples be from each school if we want the two sampling distributions to have approx the same standard deviation?
take the same size samples from each school
45% of SRS of 500 adults liked voucher idea
-What is the mean of the sampling distribution of ^p, the proportion of adults in samples of 500 who favor vouchers?
Mp^ = p = .45
45% of SRS of 500 adults liked voucher idea
-What is the standard deviation of p^?
Op^ = sq: (p)(1-p) / n
-sq (.45)(.55) / 500 = .022
45% of SRS of 500 adults liked voucher idea
-Check that you can use normal approximation for distribution of p^
np > 10 and n(1-p) > O
np = 500(.45) = 225
n(1-p) = 500(.55) = 275
45% of SRS of 500 adults liked voucher idea
-What is the probability that more than half are in favor?
Z > .5 - .45 / .22
(standard deviation of ^p: sq: (p)(1-p) / n
-sq (.45)(.55) / 500 = .022
statistical inference
use information form a sample to draw conclusions about a wider population
-different samples yield different statistics: need to be able to describe sampling distribution of statistic values
parameter vs. statistics
1) parameter: a number that describes some characteristic of the population (mean =M, SD= O, proportion= P)
2) Statistic: a number that describes some characteristic of a sample (mean = x-, SD = S, proportion = P^)
Sampling Distribution
the distribution of values taken by the statistic in all possible samples of the same size from the same population
Describing Sampling Distribution
-center: unbiased or biased (unbiased if mean of sampling distribution is equal to true value of parameter); bias if sample is not within range given (aim is off constantly; not in center)
-spread: describes variability (determined by size); high variabil
sample distribution of p^
-answers how good of an estimate statistic ^p is for parameter p
describing sample distribution of p^
-shape: sometimes can be approximated by a normal curve; depends on sample size (n) and population proportion (p)
-center: mean of the distribution is up = p
-Spread: for a specific value of p, standard deviation Op gets smaller as n gets larger
mean and SD of sample distribution of p^
(n = population, p = success, p^ is sample success)
-mean of sample p^: up = p
-SD of p^: Op = sq : (p)(1-p) / n
-ONLY IF: 10% condition is met (n)(.10) > 10
sample distribution is approx normal if
np >= 10
n(1-p) >= 10
sampling distribution of x- (sample mean)
x- is the mean of SRS of size n drawn from large population with mean u and SD O
-mean = ux= u
-SD: Ox = O/sq(N)
-ONLY IF 10% condition is met (n)(.10) > 10
-does not matter about size of population: if population distribution is normal, then so is the sa
The Central Limit Theorem
when n is large (>30) the sampling distribution of the sample mean X- is approximately normal (as sample size increases, distribution of sample means change shape: looks more like a Normal distribution and less like a population)
-allows us to use Normal