1 어떤 제품의 품질 특성을 조사하기 위하여 16개의 표본을 추출하여 측정한 결과다음과 같은 값들을 얻었다10 11 12 13 12 15 11 13 15 12 12 12 15 12 13 14 17a 특성값평균을 추정하여라b 표준오차를 구하여라c 95 신뢰구간을 구하여라 给出详细步骤请用英文回答

a) To estimate the characteristic value (mean), we need to calculate the sum of all the values and divide by the number of samples.

Sum of values = 10 + 11 + 12 + 13 + 12 + 15 + 11 + 13 + 15 + 12 + 12 + 12 + 15 + 12 + 13 + 14 + 17 = 216 Number of samples = 16

Mean = Sum of values / Number of samples Mean = 216 / 16 Mean = 13.5

Therefore, the estimated characteristic value (mean) is 13.5.

b) To calculate the (standard) error, we need to use the following formula:

Error = Standard deviation / Square root of the number of samples

First, we need to calculate the standard deviation:

Step 1: Calculate the mean (which we already did in part a) Mean = 13.5

Step 2: Calculate the deviations of each value from the mean Deviation = Value - Mean

10 - 13.5 = -3.5 11 - 13.5 = -2.5 12 - 13.5 = -1.5 13 - 13.5 = -0.5 12 - 13.5 = -1.5 15 - 13.5 = 1.5 11 - 13.5 = -2.5 13 - 13.5 = -0.5 15 - 13.5 = 1.5 12 - 13.5 = -1.5 12 - 13.5 = -1.5 12 - 13.5 = -1.5 15 - 13.5 = 1.5 12 - 13.5 = -1.5 13 - 13.5 = -0.5 14 - 13.5 = 0.5 17 - 13.5 = 3.5

Step 3: Square each deviation (-3.5)^2 = 12.25 (-2.5)^2 = 6.25 (-1.5)^2 = 2.25 (-0.5)^2 = 0.25 (-1.5)^2 = 2.25 (1.5)^2 = 2.25 (-2.5)^2 = 6.25 (-0.5)^2 = 0.25 (1.5)^2 = 2.25 (-1.5)^2 = 2.25 (-1.5)^2 = 2.25 (-1.5)^2 = 2.25 (1.5)^2 = 2.25 (-1.5)^2 = 2.25 (-0.5)^2 = 0.25 (0.5)^2 = 0.25 (3.5)^2 = 12.25

Step 4: Add up all the squared deviations 12.25 + 6.25 + 2.25 + 0.25 + 2.25 + 2.25 + 6.25 + 0.25 + 2.25 + 2.25 + 2.25 + 2.25 + 2.25 + 2.25 + 0.25 + 0.25 + 12.25 = 53

Step 5: Divide the sum of squared deviations by (number of samples - 1) Standard deviation = Square root of (53 / 15) Standard deviation = 1.8257

Now we can calculate the error:

Error = 1.8257 / Square root of 16 Error = 0.4564

Therefore, the (standard) error is 0.4564.

c) To calculate the 95% confidence interval, we need to use the following formula:

Confidence interval = Mean ± (t-value x Error)

We already know the mean and the error from parts a and b. The t-value can be found using a t-distribution table with 15 degrees of freedom (number of samples - 1) and a confidence level of 95%. The t-value for this table is 2.131, so we can substitute everything into the formula:

Confidence interval = 13.5 ± (2.131 x 0.4564)

Lower bound = 13.5 - (2.131 x 0.4564) = 12.53 Upper bound = 13.5 + (2.131 x 0.4564) = 14.47

Therefore, the 95% confidence interval is [12.53, 14.47]