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The population proportion P = 6% = 0.06 

The null hypothesis: H₀ : P ≤ 0.06 

Alternative hypothesis: H₁ : P > 0.06 

For α = 1% = 0.01. Z_α = 2.33

The test statistic is P + 2.33 (√(0.06)(0.094)/300) = 0.092 (where n = 300, P = 0.06, Q = 0.94) 

Since the calculated value is less than the table value, 0.092 < 2.33,we accept the null hypothesis H₀ . 

Hence the statement of the company can be accepted.

The level of significance is defined as the probability of rejecting a null hypothesis by the test when it is really true, which is denoted as α. That is P(Type 1 error) = α. For example, the level of significance 0.1 is related to the 90% confidence level.

A critical value is a point on the test distribution that is compared to the test statistic to determine whether to reject the null hypothesis. It depends on the level of significance. 

For example, if the confidence level is 90% then the critical value is 1.645.

(a) hypothesis

(a) population parameters

In statistical hypothesis testing, a Type f error is the rejection of a true null hypothesis. Example of Type I errors includes a test that shows a patient to have a disease when he does not have the disease, a fire alarm going on indicating a fire when there is no fire (or) an experiment indicating that medical treatment should cure a disease when in fact it does not.

A confidence interval L a type of interval estimate, computed from the statistics of the observed data, that might contain the true value of an unknown population parameter. The numbers at the upper and lower end of a confidence interval are called confidence limits. 

For example, if mean is 7.4 with a confidence interval (5.4, 9.4), then the numbers 5.4 and 9.4 are the confidence limits.

A null hypothesis is a type of hypothesis, that proposes that no statistical significance exists in a set of given observations. 

For example, let the average time to cook a specific dish is 15 minutes. The null hypothesis would be stated as “The population mean is equal to 15 minutes”, (i.e) H₀: µ = 15

A population is a set of similar items or events which is of interest to some question or experiment. A population can be specific or vague. Examples of population defined vaguely include the number of newborn babies in Tamil Nadu, a total number of tech startups in India, the average height of all exam candidates, mean weight of taxpayers in Chennai, etc. Examples of population defined specifically include a number of fans produced in a particular factory, the number of students in a class, the number of boys and girls in a tuition center, etc.

Sample size = 600 

No. of defective apples = 36 

Sample proportion p = 36/600 = 0.06 

Population proportion P = probability of defective apples = 4% = 0.04 

Q = 1 – P = 1 – 0.04 = 0.96 

The S.E for sample proportion is given by S.E

= √PQ/N 

= √(0.04)(0.96)/600 

= √0.000064 

= 0.008 

A sample is a set of data collected from a statistical population by a defined procedure. The elements of a sample are called sample size or sample points. Samples are collected and statistics are calculated from the samples so that one can make inferences from the sample to the population.

(b) More than one time

The correct answer is : (e) Error

A statistic is used to estimate the value of a population parameter. For instance, we selected a random sample of 100 students from a school with 1000 students. The average height of the sampled students would be an example of a statistic. 

Examples, sample variance, sample quartiles, sample percentiles, sample moments, etc.

A parameter is any numerical quantity that characterizes a given population or some aspect of it. This means the parameter tells us something about the whole population. 

For example, the population means µ, variance σ2, population proportion P, population correlation ρ.

An estimator is a statistic that is used to infer the value of an unknown population parameter in a statistical model. The estimator is a function of the data arid so it is also a random variable.

1. Statistical hypothesis

2. Research hypothesis 

3. Level of significance 

4. Composite hypothesis 

5. Simple hypothesis

(c) a stratified random sample

(b) sufficient

(a) Two types

The different types of sampling are 

•   simple random sampling 

•   Stratified random sampling and 

•   Systematic sampling

(i) In simple random sampling, every item of the population has an equal chance for being selected. The sampling can be done with replacement (or) without replacement. A random sampling from a finite population with replacement is equivalent to sampling from an infinite population without replacement. This technique will give useful results only if the population is homogeneous. The following are some of the methods of selecting a random sample.

(a) Use of an unbiased die or coin: If we have to choose between two alternatives, a coin is tossed and depending on the head or tail course of action is taken. A die can be employed if there are six different alternatives.

(b) Lottery sampling: Here a random sample is selected by identifying each element of the population by means of a card of a pack of uniform cards or (by writing the number on pieces of paper) and to select a required number of cards after thorough mixing of the cards.

(c) Random numbers: Random numbers are formed of ‘random digits’ and arranged in the form of a table having a number of rows and columns. Tippett’s numbers form one such table wherein 40,000 digits were selected at random from census reports and combined by groups of four into 10,000 numbers.

(ii) In stratified random sampling, a population of units is divided into L sub-populations of N₁ , N₂ , …… N_L . The sub-populations being non-overlapping and mutually exhaustive so that N = N₁ , N₂ , …… N_L . Each subpopulations is known as a stratum. If we select n₁ , n₂ , ……. n_l  items, respectively, from these strata, we get a stratified sample. If a simple random sample is taken from each stratum, the whole procedure is referred to as stratified random sampling.

(iii) Systematic sampling is a form of restricted random selection which is highly useful in surveys concerning enumerable population. In this method, every member of the population is numbered in serial order and every ith element, starting from any of the first items is chosen. For example, suppose we require a 5% sample of students from a college where there are 2000 students, we select a random number from 1 to 20. If itis 12, then our sample consists of students with numbers 12, 32, 52, 72, …… 1992.

(b) random sample

There are many ways to select a sample of 10 3-digit even numbers. From the table, start from the first number and move along the column. Select the first three digits as the number. If it is an odd number, move to the next number. The selected sample is 416, 664, 952, 748, 524, 914, 154, 340, 140, 276.

•  Systematic samples are not random samples. 

•  If N is not a multiple of n, then the sampling interval (k) cannot be an integer, thus sample selection becomes difficult.

Merits of systematic sampling are given below: 

•   This method distributes the sample more evenly over the entire listed population. 

•   The time and work are reduced much.

•  A random stratified sample is superior to a simple random sample because it ensures representation of all groups and thus it is more representative of the population that is being sampled. 

•  A stratified random sample can be kept small in size without losing its accuracy. 

•  It is easy to administer if the population under study is sub-divided.

Systematic sampling: 

In systematic sampling, randomly select the first sample from the first k units. Then every kth member, starting with the first selected sample, is included in the sample.

Systematic sampling is a commonly used technique if the complete and up-to-date list of the sampling units is available. We can arrange the items in numerical, alphabetical, geographical, or in any other order. The procedure of selecting the samples starts with selecting the first sample at random, the rest being automatically selected according to some predetermined ( pattern. A systematic sample is formed by selecting every item from the population, where k refers to the sample interval. The sampling interval can be determined by dividing the size of the population by the size of the sample to be chosen. 

That is k =N/n, where k is an integer. 

k = Sampling interval, N = Size of the population, n = Sample size.

Procedure for selection of samples by systematic sampling method :

(i) If we want to select a sample of 10 students from a class of 100 students, the sampling interval is calculated as k = N/n = 100/10 = 10 

Thus sampling interval = 10 denotes that for every 10 samples one sample has to be selected.

(ii) The first sample is selected from the first 10 (sampling interval) samples through random selection procedures. 

(iii) If the selected first random sample is 5, then the rest of the samples are automatically selected by incrementing the value of the sampling interval (k = 10)

i.e., 5, 15, 25, 35, 45, 55, 65, 75, 85, 95.

Ex: Suppose we have to select 20 items out of 6,000. The procedure is to number all the 6,000 items from 1 to 6,000. The sampling interval is calculated as k = N/n = 6000/20 = 300. Thus sampling interval = 300 denotes that for every 300 samples one sample has to be selected. The first sample is selected from the first 300 (sampling interval) samples through random selection procedures. If the selected first random sample is 50, then the rest of the samples are automatically selected by incrementing the value of the sampling interval (k = 300) ie,50, 350, 650, 950, 1250, 1550, 1850, 2150, 2450, 2750, 3050, 3350, 3650, 3950, 4250, 4550, 4850, 5150, 5450, 5750. Items bearing those numbers will be selected as samples from the population.

Sampling Errors: Errors, which arise in the normal course of investigation or enumeration on account of chance, are called sampling errors. Sampling errors are inherent in the method of sampling. They may arise accidentally without any bias or prejudice.

Sampling Errors arise primarily due to the following reasons: 

•  Faulty selection of the sample instead of the correct sample by defective sampling technique. 

•  The investigator substitutes a convenient sample if the original sample is not available while investigation. 

•  In area surveys, while dealing with borderlines it depends upon the investigator whether to include them in the sample or not. This is known as the Faulty demarcation of sampling units.

•  In simple random sampling personal bias is completely eliminated.

•  This method is economical as it saves time, money and labour.

Non-Sampling Errors:

The errors that arise due to human factors which always vary from one investigator to another in selecting, estimating or using measuring instruments( tape, scale) are called Non-Sampling errors. 

It may arise in the following ways:

•  Due to negligence and carelessness of the part of either investigator or respondents. 

•  Due to the lack of trained and qualified investigators. 

•  Due to the framing of a wrong questionnaire. 

•  Due to applying the wrong statistical measure 

•  Due to incomplete investigation and sample survey.

The correct answer is : (b) 1/N

(c) simple random sampling

The correct answer is : (a) Harper

(c) Reject H₀ when it is true 

(a) Accept H₀ when it is wrong

(d) consistent

The correct answer is : (c) σ/√n

(a) efficiency

(d) None of these

(a) Population

(b) statistic

(a) population parameter

The two branches of statistical inference are the estimation and testing of hypotheses.

(i) – (c)

(ii) – (d) 

(iii) – (a) 

(iv) – (e) 

(v) – (b)

(b) Statistic

(a) estimation

(a) a sample

(c) Standard error

Given n = 1000, \(\overline{X}\)= 119, σ = 30

S.E = σ/√n 

= 30/√1000 

= 30/31.623 = 0.9487

The alternative hypothesis is the hypothesis that is contrary to the null hypothesis and it is denoted by H. 

For example,  if H₁ : µ = 15, then the alternative hypothesis will be : H₁ : µ ≠ 15, (or) H₁ : µ < 15 (or) H₁ : µ > 15.