1. For each of the following test z- test versus the t-test versus the chi squared test. Describe wh

t-test:
t-test for one sample: It is used to determine if there is any difference between the sample mean from some hypothesized mean.
The test statistic is given below:
${\displaystyle t=\frac{\bar{\chi }-\mu }{\frac{s}{\sqrt{n}}}}$
${\displaystyle s=\frac{1}{n-1}\sum _{\{t=1\}}^n{({\chi }_1-\bar{\chi })}^2}$ is the sample standard deviation.
The distribution of the test statistic follows student’s t-distribution with (n-1) degrees of freedom. The plot of t-distribution is same as of standard normal distribution, with only difference i.e., t-distribution’s probability curve has more spread and less peak.
Two sample t test is a statistical test applied to two samples which are dependent and paired observations are taken.
Conditions for a t-test:
1. The sample size is small, n less than 30 but should not be less than 5.
2. Standard deviation is not known
3. Scale of measurement applied to the data collected follows a continuous or ordinal scale
4. When a normal distribution is assumed, level of significance or p as a criterion for acceptance is specified.
5. Standard deviations or samples are approximately equal.
Z-test:
It is a hypothesis test which is used to determine if there is a significant difference between means of two populations or to test the hypothesis about the mean of a population based on a single sample (or) to test the hypothesis about the proportion of successes in a single sample (or) the difference between the proportion of successes in two samples. This test is generally used for large samples drawn from a normal population.
It determines to what extent a data point is away from its mean of the data set, in standard deviation. The test statistic is given by:
${\displaystyle z=\frac{\bar{\chi }-{\mu }_O}{\frac{\sigma }{\sqrt{n}}}}$
where
${\displaystyle \bar{\chi }}$ is the sample mean
${\displaystyle \sigma }$ is the population standart deviation
${\displaystyle \mu }$ is the population mean
${\displaystyle n}$ is the sample size
Condition for z-test:
1. Sample size is large i.e., n>30
2. Sampling distribution must be normal or approximately normal
3. Population standard deviation is known.
Chi-square test:
This test is used for qualitative (categorical) data. Its advantage over Z test is, it can be applied for smaller samples as well as large samples. There are two types of chi-square tests:
Chi-square test for goodness of fit: It determines if a sample data matches a population
Chi-square test for Independence: It is used to test whether distributions of categorical variables differ from each other. The chi-square statistic is given as:
${\displaystyle {\chi }^2=\sum \frac{{(O_i-E_i)}^2}{E_i}}$
Where
${\displaystyle O_i}$ is the observed frequency
${\displaystyle E_i}$ is the expected frequency
The test statistic follows chi-square distribution with (r-1) (c-1) degrees of freedom
Conditions for chi-square test:
1. Individual observations of sample are independent.
2. The data should be expressed in original units, in the form of frequencies, not in ratio or percentages form
3. The total number of observations collected must be large (at least 10) and should be done on a random basis.
4. This test is used to draw inferences through test of hypothesis, cannot be used for estimation of parameter value.
5. Each cell in the contingency table has expected frequency of at least five.
Positive correlation is the degree to which two variables are linearly related. As the independent variable increases there is an increase in the dependent variable.
An example of positive correlation could be height and weight, taller the people, heavier are their weights. Let X be the independent variable defined as the height of a person and Y be the dependent variable representing the weight of a person. As the height increases, weight tends to increase, which is a positive correlation.Correlation between two variables does not mean that the change in one variable is the cause of the change in the values of the other variable. Correlation only shows if there is a relationship between variables.

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