## 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 Z-Statistic
• 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+8-2 => 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

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