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8 . The 99% confidence interval, because it includes all but one percent of the distribution. The 95% confidence interval will be narrower, because it excludes five percent of the distribution.

8.2: confidence interval, single population mean, standard deviation unknown, student’s t

9 . The t -distribution will have more probability in its tails (“thicker tails”) and less probability near the mean of the distribution (“shorter in the center”).

10 . Both distributions are symmetrical and centered at zero.

11 . df = n – 1 = 20 – 1 = 19

12 . You can get the t -value from a probability table or a calculator. In this case, for a t -distribution with 19 degrees of freedom, and a 95% two-sided confidence interval, the value is 2.093, i.e.,
t α 2   = 2.093 . The calculator function is invT(0.975, 19).

13 . E B M =   t α 2 ( s n ) = ( 2.093 ) ( 0.3 20 ) =   0.140
98.4 ± 0.14 = (98.26, 98.54).
The calculator function Tinterval answer is (98.26, 98.54).

14 . t α 2 = 2.861. The calculator function is invT(0.995, 19).
E B M = t α 2 ( s n ) = ( 2.861 ) ( 0.3 20 ) = 0.192
98.4 ± 0.19 = (98.21, 98.59). The calculator function Tinterval answer is (98.21, 98.59).

15 . df = n – 1 = 30 – 1 = 29. t α 2   = 2.045
E B M =   z t ( s n ) = ( 2.045 ) ( 0.3 30 ) =   0.112
98.4 ± 0.11 = (98.29, 98.51). The calculator function Tinterval answer is (98.29, 98.51).

8.3: confidence interval for a population proportion

16 . p = 280 500 = 0.56
q = 1 p = 1 0.56 = 0.44
s = p q n = 0.56 ( 0.44 ) 500 = 0.0222

17 . Because you are using the normal approximation to the binomial, z α 2 = 1.96 .
Calculate the error bound for the population ( EBP ):
E B P =   z a 2 p q n = 1.96 ( 0.222 ) = 0.0435
Calculate the 95% confidence interval:
0.56 ± 0.0435 = (0.5165, 0.6035).
The calculator function 1-PropZint answer is (0.5165, 0.6035).

18 . z α 2 = 1.64
E B P =   z a 2 p q n = 1.64 ( 0.0222 ) = 0.0364
0.56 ± 0.03 = (0.5236, 0.5964). The calculator function 1-PropZint answer is (0.5235, 0.5965)

19 . z α 2 = 2.58
E B P =   z a 2 p q n = 2.58 ( 0.0222 ) =   0.0573
0.56 ± 0.05 = (0.5127, 0.6173).
The calculator function 1-PropZint answer is (0.5028, 0.6172).

20 . EBP = 0.04 (because 4% = 0.04)
z α 2 = 1.96 for a 95% confidence interval
n =   z 2 p q E B P 2 =   1.96 2 ( 0.5 ) ( 0.5 ) 0.04 2 =   0.9604 0.0016 = 600.25
You need 601 subjects (rounding upward from 600.25).

21 . n =   n 2 p q E B P 2 =   1.96 2 ( 0.6 ) ( 0.4 ) 0.04 2 =   0.9220 0.0016 = 576.24
You need 577 subjects (rounding upward from 576.24).

22 . n =   n 2 p q E B P 2 =   1.96 2 ( 0.5 ) ( 0.5 ) 0.03 2 =   0.9604 0.0009 = 1067.11
You need 1,068 subjects (rounding upward from 1,067.11).

9.1: null and alternate hypotheses

23 . H 0 : p = 0.58
H a : p ≠ 0.58

24 . H 0 : p ≥ 0.58
H a : p <0.58

25 . H 0 : μ ≥ $268,000
H a : μ <$268,000

26 . H a : μ ≠ 107

27 . H a : p ≥ 0.25

9.2: outcomes and the type i and type ii errors

28 . a Type I error

29 . a Type II error

30 . Power = 1 – β = 1 – P (Type II error).

31 . The null hypothesis is that the patient does not have cancer. A Type I error would be detecting cancer when it is not present. A Type II error would be not detecting cancer when it is present. A Type II error is more serious, because failure to detect cancer could keep a patient from receiving appropriate treatment.

32 . The screening test has a ten percent probability of a Type I error, meaning that ten percent of the time, it will detect TB when it is not present.

Questions & Answers

how can l calculate G. M from the following size 125 133 141 173 182 frequency 7 5 4 1 3
Stancy Reply
how they find mean population
Joy Reply
parts of statistics
Edwin Reply
what is a mean?
Onele Reply
given the sequence 128,64,32 find the 12th term of the sequence
Shehu Reply
12th number is 0.0625
Thangarajan
why do we use summation notation to represent set of observations
MICHAEL Reply
what is the potential outlier ?
Anik Reply
A pharmaceutical company claims that their pain reliever capsule is 70% effective. But a clinical test on this capsule showed 65 out of 100 effectiveness
jelly Reply
Part of statistics
charls Reply
how to find mean population
Dawit Reply
what is data value
Ravneet Reply
what is relative frequency
Adeyemi Reply
liner regression analysis
Swathy Reply
Proper definition of outlier?
Sumbal Reply
Extraordinary observation (too distant, high, low etc)
Petr Reply

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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