Assuming CLT applies for the sample data given by 54, 56, 59, 59, 65, 66, 67, 68, 69, 76, 84, 86, 89, 94, 101 Find a 90% CI to three decimals for the mean
What is the margin for error
The margin of error (MOE) is calculated by multiplying the standard error (SE) by the critical value. The critical value is obtained from the z-table corresponding to the desired level of confidence.
To find the standard error, we first need to calculate the sample mean and the sample standard deviation.
Sample mean (x̄): (54 + 56 + 59 + 59 + 65 + 66 + 67 + 68 + 69 + 76 + 84 + 86 + 89 + 94 + 101) / 15 = 73.2
Sample standard deviation (s): √[(∑(x - x̄)²) / (n - 1)]
Plugging in the values: √[(54-73.2)² + (56-73.2)² + (59-73.2)² + (59-73.2)² + (65-73.2)² + (66-73.2)² + (67-73.2)² + (68-73.2)² + (69-73.2)² + (76-73.2)² + (84-73.2)² + (86-73.2)² + (89-73.2)² + (94-73.2)² + (101-73.2)²] / (15 - 1)
Calculating the values inside the square root: 551.6 / 14
Taking the square root: √39.4 ≈ 6.275
Now, the standard error (SE) is obtained by dividing the sample standard deviation by the square root of the sample size:
SE: 6.275 / √15 ≈ 1.617
The critical value corresponding to a 90% confidence level is found by subtracting (1 - 0.90) / 2 from 1 and looking up that value in the z-table. For a 90% confidence level, the critical value is approximately 1.645.
Finally, the margin of error (MOE) is calculated by multiplying the standard error by the critical value:
MOE: 1.617 * 1.645 ≈ 2.660
Therefore, the margin of error for the mean is approximately 2.660.