✰ Frequentist Inference:

Frequentist inference is the process of deriving conclusions about an underlying distribution via the observation of data.

we'll be writing Python codes to apply the following statistical concepts:

• The z-statistic
• The t-statistic
• The difference and relationship between the two
• The Central Limit Theorem, including its assumptions and consequences: to apply frequentist techniques to answer questions that pertain to very non-normally distributed data from the real world.
• How to estimate the population mean and standard deviation from a sample
• The concept of a sampling distribution of a test statistic, particularly for the mean
• Identifying the correct way to estimate the standard deviation of a population (the population parameter) from a sample
• How to combine these concepts to calculate a confidence interval
• Calculating critical values, confidence intervals and p-values
• Hypothesis testing: forming a hypothesis and framing the null and alternative hypotheses.

There are two parts to the project:

• In part A, we’ll highlight the Pythonic implementation of the concepts underlying frequentist inference.
• In Part B, we’ll apply those implementations to a real-world scenario.

1. Sample & Population:

In general, the sample mean we calculate will not be equal to the population mean. A consequence of this is that the sum of squares of the deviations from the population mean will be bigger than the sum of squares of the deviations from the sample mean. In other words, the sum of squares of the deviations from the sample mean is too small to give an unbiased estimate of the population variance. An example of this effect is given here. Scaling our estimate of the variance by the factor 𝑛/(𝑛−1) gives an unbiased estimator of the population variance. This factor is known as Bessel's correction. The consequence of this is that the 𝑛 in the denominator is replaced by 𝑛−1 .

2. Standard Normal Distribution:

3. Margin of Error:

A margin of error tells you how many percentage points your results will differ from the real population value. For example, a 95% confidence interval with a 4 percent margin of error means that your statistic will be within 4 percentage points of the real population value 95% of the time.

More technically, the margin of error is the range of values below and above the sample statistic in a confidence interval. The confidence interval is a way to show what the uncertainty is with a certain statistic (i.e. from a poll or survey). For example, a poll might state that there is a 98% confidence interval of 4.88 and 5.26. That means if the poll is repeated using the same techniques, 98% of the time the true population parameter (parameter vs. statistic) will fall within the interval estimates (i.e. between 4.88 and 5.26) 98% of the time.

The margin of error can be calculated after finding the values of:

1. Critical value of the sample
2. Standard Error of the sample

Margin of Error = Critical value * Standard Error

Critical value of the sample can be calculated using below: confidence level= 95%

1. Let's calculate alpha: by subtract 95% (confidence level) from 100%. That gives us 5% (0.05). This alpha level is the area in both of these tails. What we are interested in is this critical value.
2. Divide Alpha by 2 and this gives (0.025), because we're only interested in the area for one tail, not the area for both tails. So this is the area in one tail, this 5% alpha level.
3. Subtract 0.025 from 1 (1-0.025 = 0.975): The reason we did this step is we're subtracting this area from 1 and that gives us all of this area to the left, and that means we can look up this critical value here in the z table.
4. By looking at the "z table" (as shown below)for an area of 97.5%. My area of 97.5%, that's .975 is going to fall in between: x-axis (0.05 - 0.06 = 0.055) and y-axis (1.9)
5. So the critical value is going to be 1.955 (Right) and the critical value is going to be -1.955 (left).

Standard Error of the sample can be calculated using below:

Standard Error(Sample) = 𝜎 / sqrt(n)

4. Confidence Interval :

Confidence Interval of the mean, can be calculated using: population_mean ± margin_error

5. p-value :

The p-value is the probability of obtaining the observed results of a test, assuming that the null hypothesis is correct; a smaller p-value means that there is stronger evidence in favor of the alternative hypothesis.

6. Statistical Power:

‘Statistical power’ refers to the power of a binary hypothesis, which is the probability that the test rejects the null hypothesis given that the alternative hypothesis is true.

7. The Statistical Significance:

hypothesis testing to determine statistical significance:

• State the null hypothesis and alternative hypothesis.
• Calculate the p-value, the probability of obtaining the observed results of a test assuming that the null hypothesis is true.
• Set the level of the significance (alpha) and if the p-value is less than the alpha, you would reject the null — in other words, the result is statistically significant.