A statistic instructor randomly selected four bags of oranges, each bag labeled 10 pounds, and weighed the bags.They weighed 10.4, 10.9 , 10.8, and 9.9 pounds.assume that the distribution of the weights is normal. Find a 95% confidence interval for the mean weight of all bags of oranges.

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To find the 95% confidence interval for the mean weight of all bags of oranges, we can use the formula:

Confidence Interval = Sample Mean ± Margin of Error

First, let's find the sample mean:

Sample Mean = (10.4 + 10.9 + 10.8 + 9.9) / 4 = 10.5 pounds

Next, we need to calculate the standard deviation of the sample. Since we only have a small sample size (n = 4), we need to use the sample standard deviation (s) rather than the population standard deviation (σ).

Step 1: Calculate the sample variance (s^2):
s^2 = [(10.4 - 10.5)^2 + (10.9 - 10.5)^2 + (10.8 - 10.5)^2 + (9.9 - 10.5)^2] / (4 - 1)
s^2 = [0.01 + 0.16 + 0.09 + 0.36] / 3
s^2 = 0.1867

Step 2: Calculate the sample standard deviation (s):
s = √s^2 = √0.1867 ≈ 0.432 pounds

Next, we need to calculate the margin of error. The formula for the margin of error is:

Margin of Error = (Critical Value) x (Standard Deviation / √Sample Size)

Since we want a 95% confidence interval, the critical value is based on a t-distribution with (n-1) degrees of freedom. With n = 4, our degrees of freedom is 3. Using a t-table, the critical value for a 95% confidence interval with 3 degrees of freedom is approximately 3.182.

Margin of Error = 3.182 x (0.432 / √4)
Margin of Error ≈ 2.118 pounds

Finally, we can calculate the confidence interval:

Confidence Interval = Sample Mean ± Margin of Error
Confidence Interval = 10.5 ± 2.118
Confidence Interval ≈ (8.382, 12.618)

Therefore, the 95% confidence interval for the mean weight of all bags of oranges is approximately (8.382, 12.618) pounds.

To find a 95% confidence interval for the mean weight of all bags of oranges, we can use the formula:

Confidence Interval = sample mean ± (Z * standard deviation / √n)

where:
- sample mean is the average weight of the bags of oranges
- Z is the Z-score corresponding to the desired confidence level (95% in this case)
- standard deviation is the standard deviation of the weights
- n is the number of bags in the sample.

To calculate the confidence interval, we need to find the sample mean, standard deviation, and Z-score.

Step 1: Calculate the sample mean
Add up all the weights and divide by the number of bags:
Sample mean = (10.4 + 10.9 + 10.8 + 9.9) / 4 = 10.5 pounds

Step 2: Calculate the standard deviation
Subtract the sample mean from each weight, square the differences, sum them up, divide by n-1, and take the square root:
Standard deviation = √ [((10.4 - 10.5)^2 + (10.9 - 10.5)^2 + (10.8 - 10.5)^2 + (9.9 - 10.5)^2) / (4-1)] = √ [0.42 + 0.09 + 0.09 + 0.42] / 3 = √ [0.51] ≈ 0.71 pounds

Step 3: Find the Z-score for a 95% confidence level
The Z-score for a 95% confidence level is 1.96. You can find this value from a standard normal distribution table or use statistical software.

Step 4: Calculate the confidence interval
Confidence Interval = 10.5 ± (1.96 * 0.71 / √4) = 10.5 ± 0.71

Therefore, the 95% confidence interval for the mean weight of all bags of oranges is approximately (9.79, 11.21) pounds.

Find the mean first = sum of scores/number of scores

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

95% = mean ± 1.96 SEm

SEm = SD/√n

I'll let you do the calculations.