Hypothesis testing MCQ Quiz in தமிழ் - Objective Question with Answer for Hypothesis testing - இலவச PDF ஐப் பதிவிறக்கவும்

The correct answer is Paul M Sweezy

Key Points

• ​The kinked demand curve illustrates the interdependence of firms in an oligopoly market.
• American economist Sweezy came up with the kinked demand curve hypothesis to explain the reason behind this price rigidity under oligopoly.
• According to the kinked demand curve hypothesis, the demand curve facing an oligopolist has a kink at the level of the prevailing price.
• This kink exists because of two reasons:

1. The segment above the prevailing price level is highly elastic.
2. The segment below the prevailing price level is inelastic.

Goodness of Fit of a distribution is tested by Chi-square Test.

Key Points

•  A chi-squared test (also chi-square or χ2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large.
• In simpler terms, this test is primarily used to examine whether two categorical variables (two dimensions of the contingency table) are independent in influencing the test statistic (values within the table).
• The test is valid when the test statistic is chi-squared distributed under the null hypothesis, specifically Pearson's chi-squared test and variants thereof.
• Pearson's chi-squared test is used to determine whether there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table. For contingency tables with smaller sample sizes, a Fisher's exact test is used instead.

The correct answer is 16.

Key Points The degrees of freedom for a chi-square test statistic, when testing for independence in a contingency table with five rows and five columns, would be 16.

The degrees of freedom for a chi-square test of independence is calculated as follows:

(number of rows - 1) * (number of columns - 1)
In your example, there are five rows and five columns, so the degrees of freedom would be:

(5 - 1) * (5 - 1) = 4 * 4 = 16
Therefore, the correct answer is 16. 

Null hypothesis:

• A null hypothesis, also known as conjecture is a type of hypothesis that is used in statistics that proposes that there is no difference between certain characteristics of a population.
• The alternative hypothesis proposes that there is a difference. The alternative hypothesis is a direct contradiction of the null hypothesis
• A crucial step in null hypothesis testing is finding the likelihood of the sample result if the null hypothesis were true. This probability is called the p-value.
  • A low p-value means that the sample result would be unlikely if the null hypothesis were true and leads to the rejection of the null hypothesis.
  • A high p-value means that t he sample result would be likely if the null hypothesis were true and leads to the retention of the null hypothesis.
• In the context of testing proportions, the null hypothesis typically assumes that the proportions in two or more groups are equal, and any differences observed in the proportions are due to chance.
• If the null hypothesis is accepted, it means that the observed differences in proportions are not statistically significant, and any variation can be attributed to chance.
• Conversely, if the null hypothesis is rejected, it means that the observed differences in proportions are unlikely to be due to chance and are likely to be statistically significant.

Therefore, when the null hypothesis is accepted, it can be assumed that proportions are equal and the differences are due to chance.

A statistical hypothesis is tested using a four-step process:

1. The first step for the analyst is to state the two hypotheses so that only one can be right
2. The next step is to formulate an analysis plan that outlines how the data will be evaluated
3. The third step is to carry out the plan and physically analyze the sample data
4. The final step is to analyze the results and either reject the null hypothesis or fail to reject it (accepting null hypothesis) by claiming that the observed differences are explainable by chance alone.

• Analysts look forward to rejecting the null hypothesis because doing so is a strong conclusion. 
• That requires strong evidence in the form of an observed difference that is too large to be explained solely by chance.
• Failing to reject the null hypothesis i.e. the results are explainable by chance alone is a weak conclusion because it allows that factors other than chance may be at work but may not be strong enough to be detectable by the statistical test used. 

The correct answer is Critical value.

Key Points The point where the null hypothesis gets rejected is called the critical value.

Here's a breakdown of the concept:

Hypothesis Testing:

It's a statistical procedure used to determine whether sample data supports or refutes a specific hypothesis about a population.
It involves two primary hypotheses:
Null Hypothesis (H0): The default assumption, stating no significant difference or relationship exists.
Alternative Hypothesis (Ha): The opposite of the null hypothesis, suggesting a significant difference or relationship.
Critical Value:

A boundary value on a statistical distribution that separates the region where you reject the null hypothesis from the region where you fail to reject it.
It's calculated based on the significance level (alpha), the type of test, and the test statistic being used.
Decision-Making:

If the test statistic falls beyond the critical value: Reject the null hypothesis, concluding there's evidence to support the alternative hypothesis.
If the test statistic falls within the critical value: Fail to reject the null hypothesis, indicating insufficient evidence to support the alternative hypothesis

The correct answer is C, D, B, A, E.

Key Points

C. State the null and alternative hypothesis:
Hypothesis testing begins with stating a null hypothesis (H0) and an alternative hypothesis (H1 or Ha). The null hypothesis typically represents the status quo or no effect, while the alternative hypothesis represents the claim or effect that is being tested.

D. State the level of significance (α):
The level of significance (α) is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. Commonly used values for α are 0.05 (5%) or 0.01 (1%). It helps determine the critical region for decision-making.

B. Establish critical or rejection region:
The critical or rejection region is a range of values that corresponds to the extreme outcomes that would lead to rejecting the null hypothesis. It is determined based on the chosen level of significance (α) and the distribution of the test statistic. The critical region is usually located in the tails of the distribution.

A. Select a suitable test statistic:
The choice of a test statistic depends on the nature of the data and the hypothesis being tested. Different types of data and hypotheses require different test statistics, such as t-test, z-test, chi-squared test, etc. The test statistic is calculated from the sample data and is used to assess whether the observed results are consistent with the null hypothesis.

E. Formulate a decision rule to evaluate the null hypothesis:
The decision rule specifies under what conditions the null hypothesis will be rejected. It is based on comparing the calculated test statistic to the critical values from the distribution. If the calculated test statistic falls within the critical region, the null hypothesis is rejected in favor of the alternative hypothesis. If the test statistic does not fall within the critical region, the null hypothesis is not rejected.

The correct answer is The probability of obtaining the observed sample data or more extreme results, assuming the null hypothesis is true.

Key Points P Value:

• The p-value in hypothesis testing represents the probability of obtaining the observed sample data or more extreme results, assuming the null hypothesis is true.
• In other words, it measures the strength of the evidence against the null hypothesis. A small p-value indicates that the observed data is unlikely to have occurred if the null hypothesis is true, leading to the rejection of the null hypothesis in favor of the alternative hypothesis. Conversely, a large p-value suggests that the observed data is likely to occur by chance alone, providing insufficient evidence to reject the null hypothesis.
• The p-value is compared to the predetermined significance level (α) to make a decision in hypothesis testing. If the p-value is less than or equal to the significance level, typically 0.05 or 0.01, the null hypothesis is rejected. If the p-value is greater than the significance level, the null hypothesis is not rejected.
• It's important to note that the p-value does not directly indicate the probability of the alternative hypothesis being true or the effect size of the observed data. It only quantifies the strength of the evidence against the null hypothesis based on the observed data and the assumed null hypothesis.

Key PointsA Student's t-test is referred to as a technique for testing the theory of the mean of a small sample taken from a population with a normally distributed distribution when the standard deviation of the provided population is unknown.

Important Points

• William Sealy Gosset, an Englishman, created the student's t-test and t distribution in the year 1908.
• Gosset worked at the Guinness brewery in Dublin and discovered that the small sample sizes he came across in his employment made it impossible to use the existing statistical procedures employing big samples.
• The t distribution gets closer to the bell shape of the typical normal distribution as sample size (and degrees of freedom) increases.
• The normal distribution is typically used for tests involving the mean of samples with a size greater than 30.

\(\text{Student t-Test} = \frac{\bar x_1-\bar x_2}{\sqrt{(s^2(\frac {1}{n_1}+{\frac{1}{n_2})})}}\)

Hence, it can be concluded that the Student's T test was given by William Sealy Gosset.

 Additional InformationTypes of T-test

1. Use a paired t-test if the groups are drawn from the same population (for instance, when assessing before and after an experimental treatment).
2. Use a two-sample t-test if the groups are from two different populations, such as people from two different cities (also known as independent t-test).
3. Use a one-sample t-test if a group is being compared against any standard value (for instance, comparing the acidity of any liquid to a neutral pH of 7).

The correct answer is (1) I and

Key Points

Hypothesis testing involves determining whether or not differences between two sample distributions may simply be attributed to chance.

As it is probabilistic, there is always a potential that you could draw the wrong conclusions.

There are possibilities for having two types of error: type I and type II. 

1) Type 1 error: It occurs when true null hypothesis gets rejected.

2) Type 2 error: It occurs when false null hypothesis gets accepted.

● Where the region of rejection is on both sides of the sampling distribution, it is called a two-tailed test and where it is in one side of the sampling distribution, it is called one-tailed.

● Level of significance is the probability of rejecting the null hypothesis when it is true. The critical value of z statistic for two-tailed test at 5% level of significance is 1.96.

Hence, Statement I & V are correct and Statement II, III & IV are incorrect.

Therefore, the method used to compare the means of more than two populations simultaneously is called Analysis of Variance.