confidence intervals

Question 1

what is an estimator of a population parameter?

Answer 1

An estimator of a population parameter is a random variable that uses sample information to provide an approximation of this unknown parameter

Question 2

what is an estimate?

Answer 2

A specific value of that random variable is called an estimate

Question 3

what is a point estimator?

Answer 3

A point estimator 𝜃̂ is an unbiased estimator of the parameter if its expected value (or mean) is equal to that parameter:
𝐸(𝜃̂ )=𝜇_𝜃̂ =𝜃

Question 4

what are examples of point estimators?

Answer 4

The sample mean is an unbiased estimator of the population mean,
– The sample proportion is an unbiased estimator of the population proportion,

Question 5

what is the bias?

Answer 5

The bias in is defined as the difference between its expected value and 𝜃
𝐵𝑖𝑎𝑠(𝜃̂ )=𝐸(𝜃̂ )−𝜃

Question 6

what is the bias of an unbiased estimator?

Answer 6

0

Question 7

what is the most efficient estimator?

Answer 7

The most efficient estimator is the unbiased estimator with the smallest variance

Question 8

what is the relative efficiency of 𝜃̂_1 with respect to 𝜃̂_2?

Answer 8

The relative efficiency of 𝜃̂_1 with respect to 𝜃̂_2 is the ratio of their variances:
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦= 𝑉𝑎𝑟(𝜃̂_2 )/𝑉𝑎𝑟(𝜃̂_1 )

Question 9

when is 𝜃̂_1 is said to be more efficient than 𝜃̂_2?

Answer 9

𝜃̂_1 is said to be more efficient than 𝜃̂_2 if 𝑉𝑎𝑟(𝜃̂_1 )<𝑉𝑎𝑟(𝜃̂_2 )

Question 10

what is the general form for all confidence intervals?

Answer 10

The general form for all confidence intervals is:
𝑃𝑜𝑖𝑛𝑡 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒±𝑀𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝐸𝑟𝑟𝑜𝑟

Question 11

what are the necessary assumptions to find the confidence interval estimate for the mean when varience is known?

Answer 11

Population variance 𝜎^2 is known
Population is normally distributed or, if population is not normal, sample is large

Question 12

what is the formula for the confidence interval estiamate for the mean when the population variance is known?

Answer 12

𝑥̅± [𝑧_(𝛼∕2) 𝜎/√𝑛]
where 𝑧_(𝛼∕2) is the value of the standard normal distribution, above which lies 100(𝛼∕2)% of the distribution

Question 13

what is the formula for the marginal of error?

Answer 13

The margin of error, 𝑀𝐸= [𝑧_(𝛼∕2) 𝜎/√𝑛]

Question 14

when does the margin of error fall?

Answer 14

𝑀𝐸=𝑧_(𝛼∕2) 𝜎/√𝑛
1. The population standard deviation decreases (𝜎↓)
2. The sample size increases (𝑛↑)
3. The confidence level decreases (𝛼↑)

Question 15

where do the intervals extend to ?

Answer 15

UCL=𝑥̅+𝑧_(𝛼∕2) 𝜎/√𝑛
to
LCL=𝑥̅−𝑧_(𝛼∕2) 𝜎/√𝑛

Question 16

what intervals constructed contain 𝜇 and which do not?

Answer 16

100(1−𝛼)% of intervals constructed contain 𝜇;
100(𝛼)% do not

Question 17

what are the degrees of freedom (df)?

Answer 17

This is the number of observations that are free to vary after the sample mean has been calculated

Question 18

what is the difference and similiarity between the t distribution and the normal distribution?

Answer 18

𝑡 distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal

Question 19

when is the t distribution the standard normal distribution?

Answer 19

the t distribution is equal to the standard normal distribution when the degrees of freedom is equal to infinity

Question 20

what is the formula for the t distribution with (n-1) degrees of freedome

Answer 20

𝑡=(𝑥̅−𝜇)/(𝑠∕√𝑛)

Question 21

what are the necessary assumptions needed to find the confidence interval estimation of the mean when the population variance is unknown?

Answer 21

1. Population variance 𝜎^2 is unknown - We can substitute the sample standard deviation, 𝑠, for the population standard deviation, 𝜎, in the confidence interval formula used earlier but this introduces extra uncertainty, since 𝑠 is variable from sample to sample. This is why we use the 𝑡 distribution instead of the normal distribution
2. Population is normally distributed or, if population is not normal, sample is large

Question 22

what is the formula for the confidence interval estimate when the population varience is unknown?

Answer 22

𝑥̅±𝑡_(𝑛−1,𝛼∕2) 𝑠/√𝑛

where 𝑡_(𝑛−1,𝛼∕2) is the relevant value of the 𝑡 distribution with 𝑛−1 df

Question 23

when is the sample proportion approximately normal?

Answer 23

the distribution of the sample proportion is approximately normal if 𝑛𝑝(1−𝑝)>5, with standard deviation:
𝜎_𝑝̂ =√(𝑝(1−𝑝)/𝑛)

Question 24

what is the confidence interval for the population proportion?

Answer 24

𝑝̂±𝑧_(𝛼∕2) √((𝑝̂(1−𝑝̂ ))/𝑛)
where
𝑧_(𝛼∕2) is the standard normal value for the level of confidence desired
𝑝̂ is the sample proportion
𝑛 is the sample size

Question 25

what is an example of a confidence intervals with population means and a dependent sample?

Answer 25

the same group before vs after the treatment

Question 26

what is an example of a confidence intervals with population means and a independent sample?

Answer 26

group 1 vs an independent group 2

Question 27

what is an example of confidence intervals with population proportions?

Answer 27

proportions of 1 vs proportion of 2

Question 28

what is the necessary assumption for finding the difference in means between dependent samples?

Answer 28

both populations are normally distributed

Question 29

what is the formula to find the mean difference between two dependent samples?

Answer 29

𝜇_𝑑=𝜇_𝑥−𝜇_𝑦

Question 30

what is the point estimate for the population mean diference?

Answer 30

𝑑̅= (∑(𝑥−𝑦 ) )/𝑛

Question 31

what is the sample standard deviation of d?

Answer 31

𝑠_𝑑=√((∑(𝑑_𝑖−𝑑̅ )^2 )/(𝑛−1))

Question 32

what is the formula for the confidence interval for the difference between the two population means?

Answer 32

𝑑̅±[𝑡_(𝑛−1,𝛼∕2) x 𝑠_𝑑/√𝑛]

Question 33

what is the point estimate for the difference betweent the two sample means in an independent sample?

Answer 33

𝑥̅ − 𝑦̅

Question 34

what is the formula for the confidence interval for the difference in sample means in an independent sample when varience of both samples are known?

Answer 34

(𝑥̅−𝑦̅ ) ± 𝑧_(𝛼∕2) √((𝜎_𝑥^2)/𝑛_𝑥 +(𝜎_𝑦^2)/𝑛_𝑦 )

Question 35

what is the formula for the varience of the difference in means when both variences are known and the samples are independent?

Answer 35

𝜎_(𝑥̅−𝑦̅)^2=(𝜎_𝑥^2)/𝑛_𝑥 + (𝜎_𝑦^2)/𝑛_𝑦