Question

1. Define population and sample, identify examples of each and be able to tell the difference between them.

2. How to use the Central Limit Theorem and describe the distribution of the sample mean for a normal population, giving the mean and standard deviation of mu.

3.How to apply the Central Limit Theorem and describe the distribution of the sample mean for a non-normal population, giving the mean and standard deviation of mu.

4.How to use the 68-95-99.7 Empirical rule-of-thumb and the Central Limit Theorem to find the percentage (or probability) of an event above or below a given point or between two points.

5. How to calculate and interpret z-scores.

6. How to use the z-value to calculate and interpret the area under a normal curve.

7.How to use the area under a normal curve to calculate the corresponding z-scores.

8.How to use the 68-95-99.7 Empirical rule-of-thumb to provide an estimate of a 68, 95 or 99.7 confidence interval for both means mu and proportions p.  

9.How to use the 68-95-99.7 Empirical rule-of-thumb to provide an estimate of the results of a test of hypotheses for both mu and p.

Answer 1

Solution1:

the entire group of individuals to be studied is called the population.An individual is a person or object that is a member of the population being studied.

A sample is a subset of the population that is being studied.

Ex: study says 15% of US are left handed.

A random sample of 80 indviiduals found 30 are lefthanded.

proportion is p=0.15

sample is sample proportion=p^=x/n=30/80=0.375

2. How to use the Central Limit Theorem and describe the distribution of the sample mean for a normal population, giving the mean and standard deviation of mu.

For n>30 large samples from population ,distriibution of sample means follows normal dsitribution with sample mean =xbar=mu

and sample standard devaition=s=sigma/sqrt(n).

if the sample size is large,then the sampling distribution of the mean is normal even if the original population is not normal.

if the parent population is normal,the sampling distribution is also normal even for small n.

3.How to apply the Central Limit Theorem and describe the distribution of the sample mean for a non-normal population, giving the mean and standard deviation of mu

for a non-normal population,(n>30) the shape of the distribution of xbar is approximately normal when n is large.

4.How to use the 68-95-99.7 Empirical rule-of-thumb and the Central Limit Theorem to find the percentage (or probability) of an event above or below a given point or between two points

According to empirical rule,68% of the observations lies within one standard deviation of the mean

P(x-sigma<X<x+sigma)=0.68

95% of the observations lies within two standard deviation of the mean

P(x-2*sigma<X<x+2*sigma)=0.95

99.7% of the observations lies within three standard deviation of the mean

P(x-3*sigma<X<x+3*sigma)=0.997

5. How to calculate and interpret z-scores.

z=x-mean/sd

Find sample mean,xbar

Find sample standard deviation,s

let us say a student got math score as 75,sample mean=50 and sample sd=25

for score=75 mean(score)=50 and sd(score)=25

zscore=75-50/25

=25/25

z=1

positive score mean it is 1 standard deviation above mean

negative z score means it is standard deviation below mean.