Example of Confidence Interval Calculation

Example 1: We gathered data from 50 randomly selected surveys and calculated that the average completion time was 8 minutes, with a standard deviation of 2 minutes. Find 95% confidence interval.

Solution:

To find the 95% confidence interval for the average completion time, we followed these steps:

Step 1: Sample Statistics:

• Number of surveys: 50
• Average time: 8 minutes
• Standard deviation: 2 minutes

Step 2: Confidence Level: We chose a 95% confidence level, corresponding to a z-score of about 1.96.

Step 3: Margin of Error:

• Margin of Error = 1.96 × (2 / √50) ≈ 0.557

Step 4: Confidence Interval:

• Lower limit = 8 – 0.557 ≈ 7.443 minutes
• Upper limit = 8 + 0.557 ≈ 8.557 minutes

Therefore, the 95% confidence interval for the average survey completion time is approximately 7.443 to 8.557 minutes.

Step 5: Interpretation: We can be 95% confident that the true average completion time for all surveys lies between 7.443 and 8.557 minutes.

Confidence Intervals for Population Mean and Proportion(Example)

Example 2: Estimate the average height of students in a school, out of random sample of 50 students.

Solution:

Step 1: Sample Statistics:

• Average Height: 165 cm
• Standard Deviation: 7 cm
• Sample Size: 50 students

Step 2: Confidence Level: We used a 95% confidence level, which has a corresponding z-score of about 1.96.

Step 3: Margin of Error:

• Margin of Error = 1.96 × (7 / √50) ≈ 1.94 cm

Step 4: Confidence Interval:

• Lower Limit = 165 – 1.94 = 163.06 cm
• Upper Limit = 165 + 1.94 = 166.94 cm

So, we are 95% confident that the true average height of all students in the school is between 163.06 cm and 166.94 cm.

Example 3: Estimate the proportion of smartphone users in a city, when 400 people are surveyed.

Solution:

Step 1: Survey Results:

• Number of Smartphone Users: 280
• Total Surveyed: 400
• Proportion of Smartphone Users: 280 / 400 = 0.7

Step 2: Confidence Level: We used a 90% confidence level, with a corresponding z-score of about 1.645.

Step 3: Margin of Error:

• Margin of Error = 1.645 × √[(0.7 × (1 – 0.7)) / 400] ≈ 0.041

Step 4: Confidence Interval:

• Lower Limit = 0.7 – 0.041 = 0.659
• Upper Limit = 0.7 + 0.041 = 0.741

Therefore, we are 90% confident that the true proportion of smartphone users in the city is between 0.659 and 0.741.

Confidence intervals for population mean estimate the range within which the true mean lies, based on sample data. For proportions, they estimate the range within which the true population proportion lies. Both intervals reflect statistical certainty about the estimates.

This article explains confidence intervals, their calculation, interpretation, and applications for population means and proportions in statistics.