Probability & Statistics for Machine Learning & Data Science (22)

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Confidence Intervals and Hypothesis testing

Confidence Intervals

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Difference Between Confidence and Probability

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Confidence used here is not a probability.

The population mean is unknown and fixed, not a random value.

Confidence level is the rate of success vs failure when constructing the confidence interval.

Unknown Standard Deviation

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When we know the population standard deviation, we can generate a normal distribution around the population mean with the sampling distribution quantity and the sample point estimate.

Without knowing the population standard deviation like the previous examples, we use the sample standard deviation and construct the student’s t distribution to get the confidence interval.

Since the critical value ($z_{1-\alpha/2}$) uses normal distribution, we can’t use the confidence interval formula with the known standard deviation.

So we update the formula to have the t score instead of the z score to fix the scaling issue.

The larger number of samples means bigger degrees of freedom.

So the sample standard deviation gets closer to the population standard deviation and the t distribution will get closer to the normal distribution.

Confidence Intervals for Proportion

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We slightly update the confidence interval formula to compute the confidence interval for the proportion.

The sample mean becomes the sample proportion and the standard error changes into the probability multiplied by the complement of the probability divided by the sample size and taking the square root of the formula.

 

All the information provided is based on the Probability & Statistics for Machine Learning & Data Science | Coursera from DeepLearning.AI