Question

The 9 items of a sample have the values 45, 47, 50,52,48,47,49,53,51. Find the 95- confidence interval.

Answer 1

To find the 95% confidence interval for the sample, we first need to calculate the mean and standard deviation of the sample.

Mean (x̄) = (45 + 47 + 50 + 52 + 48 + 47 + 49 + 53 + 51) / 9
Mean (x̄) = 442 / 9
Mean (x̄) = 49.11

Standard deviation (σ) = sqrt [ Σ (xi - x̄)² / N ]
Standard deviation (σ) = sqrt [ (45-49.11)² + (47-49.11)² + (50-49.11)² + (52-49.11)² + (48-49.11)² + (47-49.11)² + (49-49.11)² + (53-49.11)² + (51-49.11)² / 8 ]
Standard deviation (σ) = sqrt [ (16.14) + (4.38) + (0.79) + (6.83) + (1.00) + (4.38) + (0.30) + (14.56) + (2.08) / 8 ]
Standard deviation (σ) = sqrt [ 50.46 / 8 ]
Standard deviation (σ) = sqrt (6.3075)
Standard deviation (σ) = 2.51

Now, to find the margin of error, we use the formula:
Margin of error = (Z * σ) / sqrt(n)
where Z is the z-score for the 95% confidence level (1.96 for a large sample size) and n is the number of values in the sample.

Margin of error = (1.96 * 2.51) / sqrt(9)
Margin of error = 4.92 / 3
Margin of error = 1.64

Finally, we can find the confidence interval by subtracting and adding the margin of error to the mean:

Lower bound = x̄ - margin of error
Lower bound = 49.11 - 1.64
Lower bound = 47.47

Upper bound = x̄ + margin of error
Upper bound = 49.11 + 1.64
Upper bound = 50.75

Therefore, the 95% confidence interval for the sample is 47.47 to 50.75.