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.
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.
To construct a 99% confidence interval for the mean time a test tube can be cracked, we can use the formula:
CI = X̄ ± Z * (σ/√n)
Where:
CI = Confidence Interval
X̄ = Sample Mean
Z = Z-Score for the desired confidence level (99% corresponds to Z = 2.576)
σ = Standard Deviation
n = Sample Size
Given:
Sample Mean (X̄) = 1230
Standard Deviation (σ) = 270
Sample Size (n) = 15
Confidence Level = 99% (Z-Score = 2.576)
Substituting the values into the formula, we have:
CI = 1230 ± 2.576 * (270/√15)
Calculating the standard error (SE):
SE = σ/√n
SE = 270/√15
SE ≈ 69.69
Now, substituting the SE into the formula:
CI = 1230 ± 2.576 * 69.69
Calculating CI:
CI = 1230 ± 179.82
The upper limit of the confidence interval is given by:
Upper Limit = X̄ + CI
Upper Limit = 1230 + 179.82
Upper Limit ≈ 1409.82
Therefore, the upper limit of the 99% confidence interval for the mean time a test tube can be cracked is approximately 1409.82.
To construct a confidence interval for the mean time a test tube can be cracked, we can use the sample mean, standard deviation, and the confidence level provided. Here's how you can calculate it:
1. Begin with the sample mean (x̄) = 1230 and standard deviation (s) = 270.
2. Determine the sample size (n). Although it's not explicitly mentioned in the question, the sample size (n) is required to calculate the confidence interval. Let's assume the sample size is 15.
3. Determine the critical value (z), which corresponds to the desired confidence level. Since the confidence level is 99%, you will use a two-tailed test with a significance level of 0.01. The critical value associated with this is z = 2.576 (you can find this value from a standard normal distribution table or using a calculator).
4. Calculate the standard error of the sample mean (SE), which represents the average difference between the sample mean and the population mean. The formula to calculate SE is: SE = s / √n.
SE = 270 / √15 ≈ 69.71
5. Calculate the margin of error (E), which represents the range of values within which the population mean (μ) is likely to lie. The formula to calculate the margin of error is: E = z * SE.
E = 2.576 * 69.71 ≈ 179.92
6. Construct the confidence interval by adding and subtracting the margin of error from the sample mean:
Confidence Interval = x̄ ± E
= 1230 ± 179.92
7. Finally, calculate the upper limit of the interval by adding the margin of error to the sample mean:
Upper Limit = x̄ + E
= 1230 + 179.92
= 1409.92
Therefore, the upper limit of the 99% confidence interval for the mean time a test tube can be cracked is approximately 1409.92.