ample of 15 test-tubes tested for number of times they cracked gave the sample mean of 1230 with a standard deviation of 270. Construct 99% confidence interval mean time a test tube can be cracked, give the upper limit of the interval. Assume normal distribution.

Answer 1

To construct the 99% confidence interval for the mean number of times a test-tube can be cracked, we can use the following formula:

Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √Sample Size)

First, we need to determine the critical value associated with a 99% confidence level. We can use a standard normal distribution table or a t-distribution table. However, since the sample size is large (n = 15), we can use the Z-distribution.

The critical value for a 99% confidence level using the Z-distribution is approximately 2.576.

Plugging in the given values:

Sample Mean (x̄) = 1230
Standard Deviation (σ) = 270
Sample Size (n) = 15
Critical Value (Z) = 2.576

Confidence Interval = 1230 ± (2.576) * (270 / √15)

Now, we can calculate the confidence interval:

Confidence Interval = 1230 ± (2.576) * (270 / √15)
Confidence Interval = 1230 ± (2.576) * (270 / 3.872)
Confidence Interval = 1230 ± (2.576) * 69.661
Confidence Interval = 1230 ± 179.357

The upper limit of the interval can be found by adding the margin of error to the sample mean:

Upper Limit = 1230 + 179.357
Upper Limit = 1409.357

Therefore, the upper limit of the 99% confidence interval for the mean number of times a test-tube can be cracked is approximately 1409.357.