Confidence Intervals of Independent Samples

Population Variance Known:
 Example: University doing research on Percentages of Management & Engineering Students
 Refer Here for the xlsx sheet
 Considerations:
 The Populations are normally distributed
 The population variances are known
 The Sample Sizes are different
 We want to find 95% confidence interval for the difference between the percentages of marks of the students from engineering & Management
 Facts about the data
 Samples are big
 Population Variances are known
 So, we will be using ZStatistic
 Now Lets use the following formulas
 Variance of the difference
 Applying this formula we got the following results
 CI for the Independent Samples with Known Population variance
 Results:
 Take aways:
 We are 95% Confident that the true mean difference between engineering and management grades will fall under this interval (9.28, 4.72)
 The Whole interval is negative => engineers are consistently getting lower percentages
 If we had calculate difference of management – engineering our CI => (4.72, 9.28)
 Exercise Calculate 99% CI

Population Variance Unknown but assumed to be equal:
 Lets take an example of Coffee prices in new york and LA and try to calculate the difference of means
 Refer Here for the dataset
 Now lets calculate the Mean and Std dev of NY and LA
 Formula: We need to calculate pooled variance
 Lets apply the formula
 Lets calculate Pooled Variance and Std Deviation
 Considerations
 Population variance is unknown
 Sample Size is Small
 So we would go with T Score
 CI Formula:
 degrees of freedom => 10+82 => 16
 We are calculating 95% confidence => t 16, 0.025 => 2.12
 The value is
 Inference:
 Coffee in new york is expensive than LA
 We are 95% confident that Coffee in NY will be expensive than LA in the interval of (0.47,0.92) dollars
 Exercise: Try to Calculate for 90% CI