Hypothesis testing is the statistical process of either retaining a claim or belief made by a person that is usually about population parameters such as mean or proportion and we seek evidence from a sample for the support of the claim.

Hypothesis testing consists of two statements called the null hypothesis and alternative hypothesis and only one of them is true.

The objective of hypothesis testing is to either reject or retain a null hypothesis. In many cases, like in regression models, one would like to reject the null hypothesis to establish a significant relationship between the dependent and the independent variables.

In Goodness of fit test, that is used for checking whether the data follows a specific distribution or not, we would like to retain the null hypothesis.

## Null and Alternate hypothesis

The null hypothesis is denoted as H_0 and it refers to the statement that there is no relationship between different groups with respect to the values of a proportion parameter.

The null hypothesis is also the claim that is assumed to be true initially and would try to retain it unless there is a piece of strong evidence against the null hypothesis.

Alternate hypothesis is denoted as H_a or H_1. It is the complement of the null hypothesis that is what the researcher believes to be true and would like to reject the null hypothesis.

## Test Statistics

Test statistics is the standardized difference between the estimated value of the parameter being tested calculated from the samples and the hypothesis values in order to establish the evidence in support of the null hypothesis.

The p-value is the conditional probability of observing the statistic value when the null hypothesis is true.

Example:

The average annual salary of Machine Learning experts is at least 100,000. The corresponding null hypothesis is H_0:\mu \le 100,000. Assume that the standard deviation of the population is known and the standard error of the sampling distribution is 5000.

The standardized distance between the estimated salary from hypothesis salary is (110,000 – 100,000)/5000 = 2. we can now find the probability of observing this statistic value from the sample if the null hypothesis is true. (\mu \le 100,000).

A large standardized distance between the estimated value and the hypothesis value will result in low p-value.

p-value corresponding to Z=2 is shown below.

The probability of observing a value of 2 and higher from a standard normal distribution is 0.002275. That is if the population mean is 1,00,000 and the standard error of the sampling distribution is 5000 then the probability of a sample mean greater than or equal to 1,10,000 is 0.02275.

The value 0.02275 is the p-value, which is the evidence in support of the statement in the null hypothesis.

## Significant value

The primary objective in hypothesis testing is to either reject or fail to reject the null hypothesis. Therefore we need criteria to take a decision.

The significance level is the criteria used for making the decision regarding the null hypothesis based on the calculated p-value. Significance level is denoted by /alpha.

The significance value \alpha is the maximum threshold for p-value. The decision to reject or retain will depend on whether the calculated p-value crosses the threshold value of \alpha or not.

## One-tailed test

One-tailed hypothesis tests are used to test for effects in only one direction. When you perform a one-tailed test, the entire significance level percentage goes into the extreme end of one tail of the distribution.

In the examples above, for an alpha value of 5%, each side of the distribution has one shaded region of 5%. When you perform a one-tailed test, you must determine whether the critical region is in the left tail or the right tail.

In a one-tailed test, you have two options for the null and alternative hypotheses, which corresponds to where you place the critical region.

You can choose either of the following sets of generic hypotheses:

H_0: \mu \le 0 → The effect is less than or equal to zero. H_A: \mu \gt 0 → The effect is greater than zero.

or

H_0: \mu \ge 0 → The effect is greater than or equal to zero. H_0: \mu \lt 0 → The effect is less than zero.

Example 1:

The claim made by and agency in Bangalore is the average disposable income of families living in Bangalore is grater than ₹ 4200 with a standard deviation of 3200.Given sample size is 40000 families and the mean was estimated to be ₹ 4250.

Assuming that the population standard deviation is ₹ 3200. Conduct an appropriate hypothesis test at 95% confidence level to check the validity by the agency.

The null and hypothesis, in this case, are given by

H_0: \mu_m \le 4,00,000 H_A: \mu_m \gt 4,00,000

\mu_m is the average disposable salary. The equality symbol is always part of the null hypothesis since we have to measure the difference between the estimated value from the sample and hypothesis value.

In this case, rejection or acceptance will depend on the direction of deviation of the estimated parameter value from the hypothesis value and since the rejection region is only on one side this is a one-tailed test.

Since population standard deviation is known, we can use the z-test.

z= \frac {\overline{x} - \mu}{\sigma / \sqrt{n}} = \frac{4250-4200}{3200/\sqrt{40000}}=3.125The corresponding z-value at \alpha=0.05 for right tailed test is approximately 1.64.

In excel, you can find this [LATEX]NORMSINV(1-\alpha)[/latex] that is NORMSINV(0.95).

Since, the calculated z-value is greater than z critical value,we reject the null hypothesis.

The p-value is 0.00088 which can be find using 1-normsdist that is 1-normdist(3.125).

In SAS, you can calculate p-value by using the cdf function.

```
data _null_;
p_value=1-cdf('normal',3.125);
put p_value=;
run;
```

### Running a one-sample t test in SAS.

PROC TTEST is used to perform hypothesis testing in SAS. You can either use PROC TTEST on the input dataset (if you have the raw data)or on summary statistics.

To use PROC TTEST on summary statistics, the statistics must be in a SAS data set that contains a character variable named **_STAT_** with values ‘**N**‘, ‘**MEAN**‘, and ‘**STD**‘

```
data SummaryStats;
infile datalines dsd truncover;
input _STAT_:$8. VALUE;
datalines;
N, 40000
MEAN, 4250
STD, 3200
;
```

```
proc ttest data=SummaryStats sides=U alpha=0.05 h0=4200;
var value;
run;
```

The VAR statement indicates that the variable to be studied, while the H_0= option specifies that the mean should be compared to 4200 the hypothesized value rather than the default of 0.

The SIDES=2 option specifies the number of sides and direction. The other options are U and L

We want to find whether the mean income is more than 4200. So, the sides =U option is given which is also the default.

If rather we have to find whether the mean income is different than 4200, we would use sides=2.

The ALPHA=0.05 option requests a significance level of 0.05 or a confidence level of 95%. The default significance level is 0.05.

### Interpreting the result of the t-test

Summary statistics are displayed at the top of the output. The sample size (N), mean, standard deviation, and standard error are displayed with the minimum and maximum values if you are using raw dataset.

The 95% confidence limits for the mean and standard deviation are in the next table.

At the bottom of the output is the degrees of freedom, statistic value, and p-value for the test.

At the 0.05 significance level, this test indicates that the mean income is more than 4200 and thus we would reject the null hypothesis. (t=3.13 and p-value=0.0009 > 0.04).

## Two-tailed test

A two-tailed test is a test of a hypothesis where the area of rejection is on both sides of the sampling distribution.

If we are using a significance level of 0.05, a two-tailed test shares alpha to test the statistical significance in one direction and another half of alpha to the other direction. This means that .025 is in each tail of the distribution of your test statistic.

In a two-tailed test, regardless of the direction of the relationship, the objective is to test for the possibility of the relationship in both directions.

We should use one tailed test, only we have a good reason to expect that the difference will be in a particular direction. A two tailed test is more conservative than one tailed test because a two tailed test takes more extreme statistics to reject the null hypotheis.

Example:

A passport office claims that the passport applications are processed within 30 days of submitting the application form and all the necessary documents. The below table shows the processing time of 40 passport applicants. The population standard deviation of the processing time is 12.5 days.

Conduct a hypotheis test at significance level \alpha=0.05 to verify the claim.

16 | 16 | 30 | 37 | 25 | 22 | 19 | 35 | 27 | 32 |

34 | 28 | 24 | 35 | 24 | 21 | 32 | 29 | 24 | 35 |

28 | 29 | 18 | 31 | 28 | 33 | 32 | 24 | 25 | 22 |

21 | 27 | 41 | 23 | 23 | 16 | 24 | 38 | 26 | 28 |

**Solution:**

Null and alternative hypothesis in this case are given by

H_0: \mu_m \ge 30 H_A: \mu \le 30

The given estimated sample mean is 27.05 days.

The value of Z-statistics is given by

z=\frac{\overline{X}-\mu}{\sigma / \sqrt{n}}The critical value of left-tailed test for \alpha=0.05 is -1.6444.

Since, the critical value is less than the Z value, we fail to reject null hypothesis.

The p-value for z = -1.4926 is 0.06777 which is greater than the value of \alpha.

Hence, there is no strong evidence against null hypothesis.