class 12 kpk all boards statistics notes chapter number 2 sampling distributions.
Sampling Distributions Class 12 Notes
Q.3) Explain the following terms.
a. Sample and Population
b. Parameter and Statistic
c. Sampling Distribution
a) Sample is a small representative part of a whole lot, whereas the whole lot or the universe is called population. With the help of a sample, we can study the whole lot (population). For example we want to check the taste of the rice put in a big pot, we will draw little rice in a spoon or cup and will taste it, if it is O.K., then we will consider that the taste of the rice of big pot is also o.k.
b) Any numerical value calculated from the population is called population parameter or simply parameter, µ and α2 etc. are the parameters. While any numerical value calculated from the samples is called statistic, mean and S.D are the statistics. Parameters are always constant but statistic is a variable because it varies from sample to sample.
c) The probability distribution formed from the values of any statistic is called the sampling distribution, for example, the sampling distribution of sample means, the sampling distribution of sample variance etc.
Q.4) a. Describe the advantages of sampling over complete enumeration.
b. Explain sampling and non-sampling errors.
c. Describe briefly the distinction between probability and nonprobability sampling.
a) The reasons for sampling are largely because a sample provides time and cost savings compared to a census of the entire population. In case of large or infinite population, the census of entire population is impossible. For example, a census of the population of all those who watched a particular play on TV. is impossible; however, the sample of 200 viewers is a relatively inexpensive way to obtain the viewer rating information. Another advantage of sampling is that a sample can often result in greater accuracy than a census. This is true only when a train interviewer is needed to collect the information. Sometimes a census is even impossible. For example, in the enumeration of wild life population: a census is not a feasible alternative, because not all the animals can be located and counted.
b) The difference between the estimates of the population parameters and the true value of the population is called sampling error.
The non- sampling errors cannot be attributed to sampling fluctuations. Such errors may arise from many different sources such as mistakes in the collection of data due to personal variations or dishonesty on the part of investigator defects in the selection of sample units.
c) Probability sampling is a sampling procedure in population unit (element) has a known probability of being included in the sample. There are several types of probability sampling. (1) Simple random sampling. (2) Stratified random sampling. (3) Cluster sampling. (4) Systematic sampling.
Whereas the non-probability sampling is the procedure of sampling, in which the samples are selected by personal judgment. In other words, the population units have an unknown probability of being included in the sample. Judgment sampling and quota sampling are the examples of a non-probability sampling.
Q.5) a. Explain the simple random sampling and stratified random sampling techniques.
b. Explain what is meant by standard error.
Simple random sampling is a type of probability sampling, in which each population unit has equal chance of being selected for the sample. In other words, this is a sample technique where all members of the population are treated equally regardless of their characteristics. This method is also called random sampling. The process for selecting simple random sampling is relatively easy for small finite population but very complicated for large and infinite populations. There are three popular methods for drawing SRS. (1) Gold-fish bowl procedure. (2) Random number tables. (3) Using a computer.
In stratified random sampling, 1st the population is divided into homogeneous groups and then simple random samples are taken from these groups. These similar characteristic groups are called stratas. Best results are obtained whenever the elements in each stratum are as much alike as possible. The advantage of stratified random sampling is that one member of a sub group is usually quite similar to the other members of the subgroup while being quite different from the members of the other subgroups. The most popular method of drawing SRS from each strata is proportional allocation.
b) The standard deviation calculated from the sampling distribution of any statistic values is called the standard error of that statistic. For example, standard error of sample means standard error of sample proportion etc.
Q.6) What is meant by the sampling distribution of sample mean
a. Describe the properties of the sampling distribution of sample mean.
b. describe the sampling distribution of the sample proportion.
The sampling distribution of sample means is the probability distribution formed from the values of all possible sample means.
· Properties of sampling distribution of sample mean are:
· Mean of all possible sample means is equal to population mean.
· Variance of all possible sample means is equal to population variance divided by sample size.
· If the sampled population is normally distributed then the sampling distribution of means will also be normal.
· If the sampled population is not normally distributed then the sampling distribution will be approximately normally distributed if the sample size is sufficiently large.
b) The probability distribution formed from the value of sample proportions is called sampling distribution of sample proportion. The sample proportion is denoted by p and P denotes the population proportion. The sampling distribution of p can be used to make inferences about a population proportion in the same way that was used the sampling distribution mean to make inference about the population mean.
Q.12) a. Describe the properties of the sampling distribution of the differences between two means.
b. What is meant by the sampling distribution of difference between proportions. Describe its important properties.
DIFFERENCE BETWEEN MEAN PROPERTIES:
The mean of the sampling distribution of the difference between mean is equal to
difference between two population means.i.e
µmean1 – mean 2=µ1 – µ2
2.Standard deviation in case of with replacement will be:
Standard deviation in case of without replacement will be:
If the sample population are normally distributed then the sampling distribution of difference between sample mean will also be normally distributed.
If the sampled population are not normally distributed then the sampling distribution of difference between sample mean will be approximately normally distributed if the sample size is sufficiently large.
The probability distribution of the difference between sample proportions is called the sampling distribution of the difference between proportions.
Properties of the sampling distribution of the difference between proportions:
1)µ proportion1 -proportion2 = p1 -p2
The sampling distribution of the difference between two sample proportions is constructed in a manner similar to the difference between two means. Independent random samples of size n1 and n2 are drawn from two populations of dichotomous variables where the proportions of observations with the character of interest in the two populations are p1 and p2, respectively.
Q.16) A market Research Bureau claims that 43% of adult men and 32% of adult women read Newspapers during duty hours. If these figures are accurate, describe the sampling distribution of the difference between sample proportions from samples of sizes 350 men and 270 women.
P1=43%=43/100=0.43 q1=1-p=1-0.43=0.57 n1=350
P2=32%=32/100=0.32 q2=1-p=1-0.32=0.68 n2=270
µ p̂1 – p̂2 =p1-p2
µ p̂1 – p̂2 = 0.43 – 0.32 =0.11
µ p̂1 – p̂2=√ (p1q1/n1 + p2q2/n2)
µ p̂1 – p̂2=0.43 x 0.57 / 350 + 0.32x 0.68/270
µ p̂1 – p̂2=0.0007+0.0008
µ p̂1 – p̂2=0.0015
Now we will take the root of the above we will have the standard deviation:
Since the sampling size is sufficiently large therefore according to central limit theorem the distribution of sampling distribution of difference of proportion is normal.
Q.17) In a study to estimate the proportion of residents in a certain city and its suburbs who favour the construction of a nuclear power plant, it is found that 52 of 100 urban residents favour the construction while only 34 of 125 sub-urban residents are in favour. Describe the sampling distribution of the difference between the sample proportions.
P1=x1/n=52/100=0.52 q1=1-p=1-0.52=0.48 n1=100
P2=x2/n=34/125=0.27 q2=1-p=1-0.27=0.73 n2=125
α2 p̂1 – p̂2= (p1q1/n1 + p2q2/n2)
α2 p̂1 – p̂2=0.52*0.48 / 100 + 0.272x 0.728/125
α2 p̂1 – p̂2=0.25/100 +0.20/125
α2 p̂1 – p̂2=0.0025 +0.002
α2 p̂1 – p̂2=0.0045
Now we will take the root of the above we will have the standard deviation:
Q.19) According to the bureau of the census, in 1990. 15% of Americans participated for at least one month in some type of government assistance program. In 1991, the figure grew to 16%. Consider the sample proportion, p1 and p2 obtained from samples of sizes 250 and 300 respectively, from 1990 and 1991. What is the standard deviation of the difference between the sample proportions? What is the approximate distribution of the sampling distribution of the difference between the sample proportions?
P1=15/100 p2=16/100 n1=250 n2=300
α p̂1 – p̂2= square root(p1q1/n1 + p2q2/n2)
Putting values in the above formula:
S.DP1 –P2=√0.15*0.85/250 + 0.16*0.84/300
S.DP1 –P2=√0.0005 + 0.0004
SINCE samples size are large enough so according to the central limit theorem the sampling distribution of the difference between the proportions will be approximately normal.