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64 . μ = n p = ( 500 ) ( 0.20 ) = 100
σ = n p q = 500 ( 0.20 ) ( 0.80 ) = 8.94

65 . Fifty percent, because in a normal distribution, half the values lie above the mean.

66 . The results of our sample were two standard deviations below the mean, suggesting it is unlikely that 20 percent of the lotto tickets are winners, as claimed by the distributor, and that the true percent of winners is lower. Applying the Empirical Rule, If that claim were true, we would expect to see a result this far below the mean only about 2.5 percent of the time.

7.1: the central limit theorem for sample means (averages)

67 . The central limit theorem states that if samples of sufficient size drawn from a population, the distribution of sample means will be normal, even if the distribution of the population is not normal.

68 . The sample size of 30 is sufficiently large in this example to apply the central limit theorem. This theorem ] states that for samples of sufficient size drawn from a population, the sampling distribution of the sample mean will approach normality, regardless of the distribution of the population from which the samples were drawn.

69 . You would not expect each sample to have a mean of 50, because of sampling variability. However, you would expect the sampling distribution of the sample means to cluster around 50, with an approximately normal distribution, so that values close to 50 are more common than values further removed from 50.

70 . X ¯ N ( 25 , 0.2 ) because X ¯ N ( μ x , σ x n )

71 . The standard deviation of the sampling distribution of the sample means can be calculated using the formula ( σ x n ) , which in this case is ( 16 50 ) . The correct value for the standard deviation of the sampling distribution of the sample means is therefore 2.26.

72 . The standard error of the mean is another name for the standard deviation of the sampling distribution of the sample mean. Given samples of size n drawn from a population with standard deviation σ x , the standard error of the mean is ( σ x n ) .

73 . X ~ N (75, 0.45)

74 . Your friend forgot to divide the standard deviation by the square root of n .

75 . z =   x ¯   μ x σ x =   76 75 4.5 = 2.2

76 . z = x ¯   μ x σ x =   74.7 75 4.5 = −0.67

77 . 75 + (1.5)(0.45) = 75.675

78 . The standard error of the mean will be larger, because you will be dividing by a smaller number. The standard error of the mean for samples of size n = 50 is:
( σ x n ) =   4.5 50 = 0.64

79 . You would expect this range to include values up to one standard deviation above or below the mean of the sample means. In this case:
70 + 9 60 = 71.16 and 70 9 60 = 68.84 so you would expect 68 percent of the sample means to be between 68.84 and 71.16.

80 . 70 + 9 100 = 70.9 and 70 9 100 = 69.1 so you would expect 68 percent of the sample means to be between 69.1 and 70.9. Note that this is a narrower interval due to the increased sample size.

7.2: the central limit theorem for sums

81 . For a random variable X , the random variable ΣX will tend to become normally distributed as the size n of the samples used to compute the sum increases.

82 . Both rules state that the distribution of a quantity (the mean or the sum) calculated on samples drawn from a population will tend to have a normal distribution, as the sample size increases, regardless of the distribution of population from which the samples are drawn.

Questions & Answers

In a survey of 100 stocks on NASDAQ, the average percent increase for the past year was 9% for NASDAQ stocks.
Noor Reply
what is primary data and it's various types of collection of data?
Xaib Reply
If a random variable X takes only two values -2 and 1 such that 2 P[X=- 2]=P[X=1]=P, then find variance of x.
Dikshita Reply
2.5
Anirban
how!
Alif
can u show the solution pls?
Punam
the mean of random variable following binomial distribution is 10. the numbers of trails are 30. What the approximate value of variance?
AMIT Reply
what is chis square
Remelyn Reply
A chi-square (χ2) statistic is a test that measures how a model compares to actual observed data.
kalim
descriptive statistics basic
Sangeetha
Hi Friends
Sangeetha
tamil people yaaraavathu irukkingala
Sangeetha
what is the median of the following set of number:3,3,3,4,5,6,7,8,8,9?
Sur Reply
3
Mbemah
5.6
Alekya
(5+6)/2=5.5555.......=5.6
Lyrical
(5+6)/2=5.5 that is the median
moniba
hi friends am new here, hope am welcome
nathan
hi friends am new here, hope am welcome
nathan
1.5
nathan
differences between Intervals and Ratio measurement scales
VIP Reply
interval is the distinct from a starting point to the peak of a certain distance
Mbemah
the taking of a certain range accurately in a divisional way
Mbemah
i need the solution for exercise sums
Siva Reply
descriptive statistics sums I need
Sangeetha
statistics discovered?
Kaleem Reply
internet
Kamranali
dev OS
Virendra
searching for motivation to learn stat
Nija
purpose of of statistics
mukesh
ya..
mukesh
tamil people yaraavathu inrukkingala
Sangeetha
Odisha ru kau bhai mane achha?
Hemanta
I want to learn my MSC by statistics who can help me ?
Abebe
@Avirup interest person
Abebe
good morning
Sangeetha
Dr good morning
mukhtar
in statistical inferences we use population parameter to infer sample statistics
anupama Reply
Yes
lukman
how do we calculate decile
Drix Reply
how do we calculate class boundaries
Syed
The lower class boundary is found by subtracting 0.5 units from the lowerclass limit and the upper class boundary is found by adding 0.5 units to the upper class limit. The difference between the upper and lowerboundaries of any class.
Ekene
The lower class boundary is found by subtracting 0.5 units from the lowerclass limit and the upper class boundary is found by adding 0.5 units to the upper class limit. The difference between the upper and lowerboundaries of any class.
Ekene
complete definition plz
Zarifa Reply
what are the advantages and disadvantages of primary and secondary data
Elisha Reply
advantage of primary data 1) specific 2) Accurate 3)ownership 4)up to date information 5) control Disadvantage of primary data 1) Expensive 2) time consuming 3) feasibility.
Xaib
the average increase for all NASDAQ stocks is the
Technical Reply
formula of poisson probability distribution
Ch 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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