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Statistics- Test 2
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Terms in this set (38)
binomial distribution formulas
mean (mu)=n*p
var (sigma ^2)=n
p
q
std dev (sigma)= sqrt(n
p
q)
range rule of thumb
max/min usual= mu+-2 sigma
uniform distribution
values are spread evenly over the range of probabilities
density curve
graph of a continuous prob. dist. MUST SATISFY
1.the total area under the curve =1
2. every point on the curve must have a vertical height>0
notation
Za denotes the z score with an area of (a) to its right
conversion formula
z=(x-mu)/sigma
sampling dist. of a statistic
the dist. of all values of the statistic when ALL possible samples of the same size (n) are taken from the same population
sample mean
target (equal) the value of the population mean.
the dist. of the sample means tends to be a normal dist.
sample variance
target the value of the population var.
the dist of the sample var. tends to be a dist. skewed right
sample proportion
target the value of the pop. proportion.
tends to be a normal dist.
proportion of successes
p^ (p hat)= X/n= (# successes/total # of observations)
p= population proportion
p^= sample proportion
Unbiased estimators
*target the population parameter
Sample means
sample variances
sample proportions
*better in estimating the population parameter
Biased Estimators
*do NOT target the population parameter
Sample medians
sample ranges
standard deviations
*the bias with stdev is relatively small in large samples so (s) is often used to estimate.
Central Limit Theorem
1.distribution of sample xbar will, as sample size increases, approach a normal dist.
2.the mean of the sample means is the population mean (mu)
3. stdev of all sample means is sigma/sqrt(n)
central limit rules
1.for sample sizes (n) larger than 30, the dist. of the sample mean can be approx. by a normal. dist.
2. if the original population is normally dist. then for any sample size (n) the means will be normally dist.
standard deviation of sample means
Normal quantile plot
x= original data value
y= corresponding z score
procedure for determining sample data is from a Normally Dist. Population.
1. construct histogram (looking for general bell shape)
2. identify outliers (reject if there is more than one)
3. generate a normal quantile plot. (are the points reasonably in a straight line)
Normal Quantile Plot (manual construction)
1. sort values lowest to highest
2. using (n) calculate 1/2n, 3/2n, 5/2n, 7/2n, 9/2n .... etc
3. find z scores corresponding to cumulative left area of those vlaues.
4. plot x (original data) and y (z scores) on graph. look for line.
Approximating a Binomial Dist. with a Normal dist.
1. np>=5, nq>=5
2. find mu=np, sigma=sqrt(npq)
3. find values +- 0.5 from x.
4. determine if x is included in the probability.
Continuity corrections
"at least" (includes 235 and above): to the right of 234.5
"more than" (doesn't includes 235): to the right of 235.5
"at most" (includes 235 and below): to the left of 235.5
"fewer than" (doesn't includes 235): to the left of 234.5
"exactly": between 234.5 and 235.5
point estimate
single value used to approximate a population parameter
sample proportion p^
is the best point estimate of the population proportion p
confidence interval
a range of values used to estimate the true value of a population parameter. (abbreviated as CI)
Confidence level
degree of confidence
confidence coefficient
is the probability 1-alpha that the confidence interval actually contains the population parameter.
common choices 90%, 95%, 99% (alpha=0.10, 0.05, 0.01)
Critical values
standard z score that distinguished between sample stats that are likely to occur and those that are unlikely.
-a z score associated with a sample proportion has a probability of alpha/2 of falling in the right tail.
common critical values
confidence level 90%- critical value 1.645
confidence level 95%- critical value 1.96
confidence level 99%- critical value 2.575
Margin of Error for proportions
E= z(α/2)Sqrt(p^q^/n)
confidence level for estimating a population proportion
p^-E < p < p^ + E
padding
round off rule for estimating confidence interval
round to 3 sig. digits
Constructing a confidence level for p
1. verify np & nq >= 5
2.find critical z values that correspond to confidence level.
3. evaluate the margin of error E=
4. find confidence interval p^ +- E
5. round to 3 sig. figs.
common polling misconception
the population size is usually not a factor in determining the reliability of a poll.
determining sample size
n=(Zα/2)^2p^q^/(E^2)
when p^ is unknown use o.5 (p^q^=.25)
round off rule for determining population size
round up the value of (n) to the next larger whole number
Finding point estimate and E from a confidence interval
p^= (upper conf. limit + lower conf. limit)/2
E= (upper conf. limit - lower conf. limit)/2
Constructing a confidence interval
1. verify requirements are satisfied
2. using n-1 degrees of freedom use A3 to find t a/2 value
3. evaluate margin of error E=
4. find the value of x̄ - E and x̄ + E
Choosing appropriate distribution
Normal (Z) dist- σ is known and n>30
use (t) dist.- σ is unknown and n>30
n<= 30 : cannot solve in this course
properties of a Chi dist
-values cannot be negative
-chi-square dist. is different with each # of degrees of freedom
-use table A-4 (represents the cumulative area to the right of the critical value)
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