# 0.2 Practice tests (1-4) and final exams  (Page 15/36)

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64 . $\mu =np=\left(500\right)\left(0.20\right)=100$
$\sigma =\sqrt{npq}=\sqrt{500\left(0.20\right)\left(0.80\right)}=8.94$

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 . $\overline{X}\sim N\left(25,0.2\right)$ because $\overline{X}\sim N\left({\mu }_{x},\frac{{\sigma }_{x}}{\sqrt{n}}\right)$

71 . The standard deviation of the sampling distribution of the sample means can be calculated using the formula $\left(\frac{{\sigma }_{x}}{\sqrt{n}}\right)$ , which in this case is $\left(\frac{16}{\sqrt{50}}\right)$ . 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 $\left(\frac{{\sigma }_{x}}{\sqrt{n}}\right)$ .

73 . X ~ N (75, 0.45)

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

75 .

76 .

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:

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:
$70+\frac{9}{\sqrt{60}}=71.16$ and $70-\frac{9}{\sqrt{60}}=68.84$ so you would expect 68 percent of the sample means to be between 68.84 and 71.16.

80 . $70+\frac{9}{\sqrt{100}}=70.9$ and $70-\frac{9}{\sqrt{100}}=69.1$ 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

Write a short note on skewness
and on kurtosis too
Hiren
What is events
who is a strong man?
Can you sir plz provide all the multiple choice questions related to Index numbers.?
about probabilty i have some questions and i want the solution
What is hypothesis?
its a scientific guess
ted
A hypothesis in a scientific context, is a testable statement about the relationship between two or more variables or a proposed explanation for some observed phenomenon. In a scientific experiment or study, the hypothesis is a brief summation of the researcher's prediction of the study's findings.
Hamzah
Which may be supported or not by the outcome. Hypothesis testing is the core of the scientific method.
Hamzah
statistics means interpretation analysis and representation of numerical data
Ramzan
To check the statment or assumption about population parameter is xalled hypothesis
Ali
hypothesis is simply an assumption
Patrick
what is the d.f we know that how to find but basically my question is what is the d.f? any concept please
Degrees of freedom aren’t easy to explain. They come up in many different contexts in statistics—some advanced and complicated. In mathematics, they're technically defined as the dimension of the domain of a random vector.
Hamzah
d.f >> Degrees of freedom aren’t easy to explain. They come up in many different contexts in statistics—some advanced and complicated. In mathematics, they're technically defined as the dimension of the domain of a random vector.
Hamzah
But we won't get into that. Because degrees of freedom are generally not something you needto understand to perform a statistical analysis—unless you’re a research statistician, or someone studying statistical theory.
Hamzah
And yet, enquiring minds want to know. So for the adventurous and the curious, here are some examples that provide a basic gist of their meaning in statistics.
Hamzah
The Freedom to Vary First, forget about statistics. Imagine you’re a fun-loving person who loves to wear hats. You couldn't care less what a degree of freedom is. You believe that variety is the spice of life Unfortunately, you have constraints. You have only 7 hats. Yet you want to wear a different
Hamzah
hat every day of the week. On the first day, you can wear any of the 7 hats. On the second day, you can choose from the 6 remaining hats, on day 3 you can choose from 5 hats, and so on.
Hamzah
When day 6 rolls around, you still have a choice between 2 hats that you haven’t worn yet that week. But after you choose your hat for day 6, you have no choice for the hat that you wear on Day 7. You must wear the one remaining hat. You had 7-1 = 6 days of “hat” freedom—in which the hat you wore
Hamzah
That’s kind of the idea behind degrees of freedom in statistics. Degrees of freedom are often broadly defined as the number of "observations" (pieces of information) in the data that are free to vary when estimating statistical parameters.
Hamzah
binomial distribution and poisson both are used to estimate the number of successes probable against the. probable failures. the difference is only that BINOMIAL dist. is for discrete data while POISSON is used for continuous data.
Salman
What do you need to understand?
Angela
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution
Hamzah
poisson distribution is also for discrete data set. The difference is when the probability of occurring an event is very little and the sample size is extra large then we use poisson distribution.
Neil
Neil yes you got it and very interested answer you gave
jamilu
How to know if the statement is 1 tail or 2 tail?
1 tail if greater than pr less than.2 tail if not equal.
Jojo
in such a case there is no sufficient information provided to develop an alternative hypothesis and we can decide between only two states i.e either the statement is EQUAL TO or NOT EQUAL TO under given conditions
Salman
for 1tail there must be certain criteria like the greater than or less than or some probability value that must be achieved to accept or reject the original hypothesis.
Salman
for example if we have null hypothesis Ho:u=25 Ha:u#25(not equal to 25) it would be two tail if we say Ho:u=25 Ha:u>or Ha:u<25 it would be consider as one tail I hope you will be understand #Coleen
Shabir
yes its true. now you have another problem. so share.
ibrar
what is z score
How to find z score through calculator
Esperanza
Different data sets will have different means and standard deviations, so values from one set cannot always be compared directly with those from another. The z-score standardizes normally distributed data sets, allowing for a proper comparison and a consistent definition of percentiles across data s
Hamzah
what are random number
how to compute the mean with a long method
there is a shortcut method for calculating mean long methid doesn't make any sense.
Neil
what are probability
Saif
probability mass function
Saif
probability density function
Saif
there are many definitions of probability. which one is, the ratio of favourable outcomes & total outcomes.
suhail
distribution used for modeling/(find probabilities) of discrete r.v. is called p.m.f
suhail
distribution used for modeling/(find probabilities) of continued r.v, called p.d.f
suhail
lets use short method using calculator.... store yo data n smply get your mean
Flavian
if 1 calorie =4.12 kj, what is the total kj value of this dish
summation of values of x1 x2 x3 ,,,,xn divided by total number n if it is with frequency its like this summation of values of x1f1+x2f2+x3f3+xnfk divided by summation of frequencies like f1+f2+f3+fk
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