Statistics- Chapter 9 Point Estimates

steps for: Given a variable that has a t distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? (Round your answers to one decimal place.) 10 df, between -2.23 and 2.23%

Step 1) Check for 10 degrees of freedom in row, see for 2.23 approximately. Step 2) Highlight its respective % for 2 tail as we are considering 2.23 and -2.23 its 2.228 approximately 2.23. It's respective probability = 0.05 for two tails. That means 2.5% each side, 5% both sides together. Step 3) Subtracting 5% from 1, it will give 95% of values lies between these 2 numbers.

the interval around p hat should encompass as

% of distribution

what is margin of error?

+-

confidence interval of 95% means the width of it is

0.03 because .5 (1-.5)=

steps for: Given a variable that has a t distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? (Round your answers to one decimal place.) 25 df, outside the interval from -2.79 to 2.79%

1. 25 degrees of fredom 2. approximately 2.79 in tables it is 2.787 3. as we need -2.79 and 2.79, respective two tailed % will be = ________. 100%-_____=______ 4. 99% of the values lies between -2.79 and 2.79 for 25 degrees of freedom

steps for: Given a variable that has a t distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? (Round your answers to one decimal place.) 24 df, to the right of 2.80%

1. check for 25 degrees of freedom 2. spot approxmiately 2.80 in a row (here it is 2.79) 3. spot its respective one tailed probability as we want 2.80 4. subtract that from 100% as a whole probability curve is 100%. 5. should get 0.5% as area to right of 2.80

how to find critical value

1. compute alpha (a):a=1-(confidence level/100) 2. find the critical probability (p star)=1-(a squared) 3. to express critical value as z score, find z score having a cumulative probability=to find critical probability (p star)

how to express the critical value as a positive stat

1. find df 2. the critical positive stat (t star) is the t statistic having df=to degrees of freedom and a cumulative probability=to critical probability (p star)

general properties for sampling distributions of p hat

1. mean of p hat=p 2. sigma p hat=square root of p (1-p)/n as long as the sample size is less than 10% of the population 3. as long as n is sufficiently large (np greater than or equal to 10 and n (1-p) greater than or equal to 10) the sampling distribution of p hat is approximately normal

the general formula requirements for a confidence interval for a population proportion p is used when

1. p hat is the sample proportion from a random sample 2. sample size n is large (n p hat greater than or equal to 10 and n (1-p hat greater than or equal to 10) 3. if the sample is selected without replacement the sample size is small relative to the population size (at most 10% of the population)

the general formula requirements for a confidence interval for a population mean mu based on a sample of size n when...

1. x bar is the sample mean from a random sample 2. the population distribution is normal, or the sample size n is large (n greater than or equal to 30) and 3. sigma, the population standard deviation, is unknown is x bar +- (t critical value) (s/square root of n)

Samples of two different types of automobiles were selected, and the actual speed for each car was determined when the speedometer registered 50 mph. The resulting 95% confidence intervals for true average actual speed were (51.7, 53.1) and (49.3, 50.5). Assuming that the two sample standard deviations are identical, which confidence interval is based on the larger sample size? (49.3, 50.5)(51.7, 53.1) Explain your reasoning.

49.3 and 50.5 because formula for obtaining 95% confidence interval is mean +- 1.96 (standard deviation/square root of n) reason for answer: as sample size increases, the width of the interval decreases

what are the 3 confidence levels?

90, 95, 99%

1.96 is connected to

95% confidence

the use of the interval p hat +- (z criticla value) square root of p hat(1-phat)/n requires

a large sample

confidence intervals

a population characteristic that is an interval of plausible values for the characteristic.

point estimates

a single number ( a statistic) based on a sample data that is used to estimate a population characteristic

choosing a statistic that is unbiased (accurate)

a statistic whose mean value is equal to the value of the population characteristic being estimated

as the number of degrees of freedom increases, the corresponding sequence of t distributions

approaches the standard normal distribution

the t distribution corresponding to any particular number of degrees of freedom is

bell shaped and centered at zero (just like the standard normal z distribution)

why is the conservative estimate for p=0.5?

by using .5 for p, we are using the largest value for P (1-P) in our calculations so .1(.9)=.09 .2(.8)=.16 .3(.7)=.21 .4(.6)=.24 .5(.5)=.25

as the number of degrees of freedom increases, the spread of the corresponding t distribution

decreases

the t critical value in confidence intervals for mu when sigma is unknown is based on

df=n-1

each degree of freedom in problem means

different positive distribution

if there is no reasonable basis for eliminating p and a preliminary study is not feasible, a conservative solution to follow is

from p (1-p) is never larger than 0.25, that is when p=95. n=p(1-p)(1.96/B0) squared --> o.5 (1-0.5)(1.96/B) squared --> 0.25 (1.96/B)squared

B is what?

it is the width of confidence interval

best way to measure a sample mean from a population

mean being x bar

what is error bound?

n

if margin of error is small,

n is got to be bigger and vice versa

the sampling distribution of p hat is approximately normal if

n is sufficiently large (np greater than or equal to 10 and n (1-p) greater than or equal to 10)

how to solve n when choosing a sample mean (b=1.96 (sigma/radical n)

n=(1.96 sigma/b)squared

confidence intervals for mean when sigma is unknown

standardize the values by t=x bar- mu (mean)/sigma/square root of n

sample variance equation

sum of (x-xbar)/n-1 squared

x bar equation

sum of x /n

how to find z values and the area between those two values when given a 95% confidence level?

take .95-1=.050 (remaining area and because of the curve symmetry. divided by 2. Find z score of .050

when approximating sampling distribution of p hat, you have to

take different samples to get p hat to get confidence intervals. Using this method of calculation, the confidence interval will not capture p 5% of the time when given 95% confidence interval.

to estimate sigma

take range and divided it by 4

width of confidence interval

the difference between the lower and upper values of the interval. numbers expressed as a decimal but is a precentage + or - with a confidence interval that is correct

by using .5 for p, we are using

the largest value for p (1-o) in our calculations to get p as biggest as possible

mu

the mean of sampling distribution of sampling proportions

the confidence level associated with a confidence interval estimate is

the success rate of the method used to construct the interval

to plug in value of p, use

value of p hat in an old study or use conservative estimate for p, use value for p to force n as large as possible

if we solve b=1.96 square root p (1-p)/n, then

we solve for n. N=phat (1- phat) (z a/2 /B)2

B

width of interval of particular size.

how are confidence intervals constructed?

with a chosen degree of confidence, the actual value of the characteristic will be between the lower and upper endpoints of the interval

sample mean

x bar. the measure of central tendency of a distribution

the bound on error estimation for a 95% confidence interval is the bound on error estimation for a 90% confidence interval is the bound on error estimation for a 99% confidence interval is

B=1.96 square root p (1-p)/n B=1.645 square root p (1-p)/n B=2.58 square root p (1-p)/n

the 1.96 coefficient changes depending on

size of distribution (90% smaller value for z*, 99% larger value for z *)

confidence intervals for mu when sigma is unknown is valid for

small values of n, smaller than 30, only when population distribution is small

each t distribution is more ___________ _______________ than the standard normal distribution

spread out

when n is in denominator, you have to

square both sides of equation

x bar has a smaller

standard deviation for a statistic to measure mean

sigma p hat

standard deviation of sampling distribution

the bound on error of estimation associated with a 95% confidence interval is

B=1.96 (sigma/square root of n)

if we were to compute 100 more confidence intervals for p from 100 different random samples, would we get the same results?

no. get different results each time you compete a number of more confidence intervals. Expect 95 of them to contain and not to contain samples

what is point estimate and proportion from sample?

p hat

the general formula for a confidence interval for a population proportion p

p hat +- (z critical value) square root p hat (1-p hat)/n

for large n, a 95% confidence interval for p is

p hat +- 1.96 square root of p (1-p)/n

what means degree of freedom?

sample of size n

choosing a sample size is looking at

sample proportions that can be converted to %

how to calculate df

sample size -1

s

sample standard deviation. means more variability and have to use a different distribution.

sq root

sampling distribution of p hat