the masses of 6 containers of Chow dog food are:
9.8
10.4
9.8
10
10.2
9.6
Find a 99% confidence interval for the mean mass of all such dog food. (Assume an approximate normal distribution). Give the lower limit.
To find the 99% confidence interval for the mean mass of all the dog food containers, we'll use the formula:
Confidence Interval = (Sample Mean) ± (Z-Score) * (Standard Deviation / √(Sample Size))
First, let's calculate the sample mean:
Sample Mean = (9.8 + 10.4 + 9.8 + 10 + 10.2 + 9.6) / 6 = 9.9667 (rounded to four decimal places)
Next, let's calculate the standard deviation (σ):
Sum of squared differences from the mean = (9.8 - 9.9667)² + (10.4 - 9.9667)² + (9.8 - 9.9667)² + (10 - 9.9667)² + (10.2 - 9.9667)² + (9.6 - 9.9667)²
= 0.02222 + 0.014379 + 0.02222 + 0.000111 + 0.000501 + 0.003518
= 0.062949 (rounded to six decimal places)
Standard Deviation = √(Sum of squared differences from the mean / (Sample Size - 1)) = √(0.062949 / (6-1)) = √(0.0125898) = 0.11202 (rounded to five decimal places)
Now, let's determine the Z-Score for a 99% confidence level.
Since we assume an approximate normal distribution, we'll use a Z-Score table or calculator to find the Z-Score for a 99% confidence level, 0.99.
The Z-Score for a 99% confidence level is approximately 2.576.
Finally, we can calculate the confidence interval:
Confidence Interval = (Sample Mean) ± (Z-Score) * (Standard Deviation / √(Sample Size))
= 9.9667 ± (2.576) * (0.11202 / √(6))
= 9.9667 ± (2.576) * (0.04577)
Lower Limit = 9.9667 - (2.576) * (0.04577)
= 9.9667 - 0.11800
≈ 9.8487
The lower limit for the 99% confidence interval for the mean mass of all such dog food is approximately 9.8487.
To calculate a 99% confidence interval for the mean mass of all the containers of Chow dog food, we can use the formula:
Confidence Interval = X̄ ± Z * (σ / √n)
Where:
X̄ is the sample mean, which is the average of the sample masses
Z is the Z-score corresponding to the desired confidence level (99% in this case)
σ is the population standard deviation (which we don't have, so we can estimate it using the sample standard deviation)
n is the sample size
First, let's find the sample mean (X̄) and sample standard deviation (s):
X̄ = (9.8 + 10.4 + 9.8 + 10 + 10.2 + 9.6) / 6 = 9.9667
Next, calculate the sample standard deviation (s):
s = sqrt([(x₁ - X̄)² + (x₂ - X̄)² + ... + (x₆ - X̄)²] / (n - 1))
= sqrt([(9.8 - 9.9667)² + (10.4 - 9.9667)² + (9.8 - 9.9667)² + (10 - 9.9667)² + (10.2 - 9.9667)² + (9.6 - 9.9667)²] / (6 -1))
= sqrt([0.00056 + 0.00014 + 0.00056 + 0.00001089 + 0.00031744 + 0.00017344] / 5)
≈ sqrt(0.0005594) ≈ 0.0237
Now, we need to find the Z-score corresponding to the 99% confidence level. Since we want to find the lower limit, we'll use the negative Z-score. Using a standard normal distribution table or calculator, we find that Z is approximately -2.62 at a 99% confidence level.
Finally, we can calculate the confidence interval:
Confidence Interval = X̄ ± Z * (σ / √n)
Confidence Interval = 9.9667 - 2.62 * (0.0237 / sqrt(6))
Confidence Interval ≈ 9.9667 - 2.62 * (0.0237 / 2.4495)
Confidence Interval ≈ 9.9667 - 0.0253
Confidence Interval ≈ 9.9414
Therefore, the lower limit of the 99% confidence interval for the mean mass of all the containers of Chow dog food is approximately 9.9414.