Sample 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 lower limit of the interval. Assume normality.

Question

Sample 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 lower limit of the interval. Assume normality.

Answer 1

To construct a 99% confidence interval for the mean time a test tube can be cracked, we will use the formula:

Confidence Interval = (sample mean) ± (critical value) * (standard deviation / sqrt(sample size))

First, we need to find the critical value corresponding to a 99% confidence level. Since the sample size is less than 30, we will use a t-distribution instead of a z-distribution.

The degrees of freedom for a t-distribution with 15 test tubes is 15 - 1 = 14.

Using a t-table or calculator, we find that the critical value for a 99% confidence level and 14 degrees of freedom is approximately 2.977.

Now, we can calculate the confidence interval:

Confidence Interval = 1230 ± (2.977) * (270 / sqrt(15))

Confidence Interval = 1230 ± 222.047

Lower Limit = 1230 - 222.047 = 1007.953

The lower limit of the 99% confidence interval for the mean time a test tube can be cracked is approximately 1007.953.

Answer 2

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

Confidence Interval = Sample Mean ± (Critical Value * Standard Error)

First, we need to calculate the critical value at 99% confidence level. Since the sample size is greater than 30, we can use the Z-distribution. The critical value for a 99% confidence level is 2.58.

Next, we need to calculate the standard error.
Standard Error = Standard Deviation / √(Sample Size)

Given that the sample mean is 1230, the standard deviation is 270, and the sample size is not mentioned, we'll assume it is 15 (as stated in the question).

Standard Error = 270 / √15
Standard Error ≈ 69.5

Now we can construct the confidence interval:
Confidence Interval = 1230 ± (2.58 * 69.5)

Lower Limit = 1230 - (2.58 * 69.5)
Lower Limit ≈ 1230 - 179.31

The lower limit of the 99% confidence interval for the mean time a test tube can be cracked is approximately 1050.69.

Answer 3

To construct a confidence interval for the population mean time a test tube can be cracked, you can use the formula:

Confidence interval = sample mean ± (critical value * standard deviation / √(sample size))

First, let's find the critical value. Since you want a 99% confidence interval, you need to find the z-score that corresponds to a 99% confidence level.

Using a standard normal distribution table or a calculator, you can find that the z-score for a 99% confidence level is approximately 2.576.

Next, plug the values into the formula:

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

To find the lower limit of the interval, subtract the result from the sample mean:

Lower limit = 1230 - (2.576 * 270 / √15)

Now, let's calculate the lower limit:

Lower limit = 1230 - (2.576 * 270 / √15)
Lower limit ≈ 1230 - (2.576 * 270 / 3.872)
Lower limit ≈ 1230 - (700.32 / 3.872)
Lower limit ≈ 1230 - 180.77
Lower limit ≈ 1049.23

Therefore, the lower limit of the 99% confidence interval for the mean time a test tube can be cracked is approximately 1049.23.