Julie was given a bucket of numbers. Unknown to Julie, the mean of the bucket is 4.9.

Julie was given given a set of instructions that read as follows.

Goal: The object of this exercise is to estimate the mean of the bucket using the procedure given below. This is a procedure for constructing a 85% Confidence Interval.

1. Take a sample, with replacement, of 79 numbers from the bucket.

2. Compute the sample mean,
𝑥
¯
.

3. Compute the number
𝑥
¯
- 8.5

4. Compute the numbers
𝑥
¯
+ 8.5

If 5. Estimate (Guess) that the mean is between
𝑥
¯
- 8.5 and
𝑥
¯
+ 8.5.

Julie completed these instructions. In step 3, she got a sample mean of
𝑥
¯
=-1.6. (All necessary assumptions for this procedure were met.)

What is the probability that the procedure outlined about will produce an interval that surrounds the mean?

c07.p061.q003

Question

Julie was given a bucket of numbers. Unknown to Julie, the mean of the bucket is 4.9.

Julie was given given a set of instructions that read as follows.

Goal: The object of this exercise is to estimate the mean of the bucket using the procedure given below. This is a procedure for constructing a 85% Confidence Interval.

1. Take a sample, with replacement, of 79 numbers from the bucket.

2. Compute the sample mean, 
𝑥
¯
.

3. Compute the number 
𝑥
¯
 - 8.5

4. Compute the numbers 
𝑥
¯
 + 8.5

If 5. Estimate (Guess) that the mean is between 
𝑥
¯
 - 8.5 and 
𝑥
¯
 + 8.5.

Julie completed these instructions. In step 3, she got a sample mean of 
𝑥
¯
=-1.6. (All necessary assumptions for this procedure were met.)

What is the probability that the procedure outlined about will produce an interval that surrounds the mean?

c07.p061.q003

BlackTom AI · Accepted Answer

This problem centers on understanding what an 85% confidence interval means in a sampling procedure.
The instruction says the goal is to construct a procedure for an 85% confidence interval around the unknown mean.
Optionally, think of the interval as x̄ − 8.5 to x̄ + 8.5 constructed from a sample of size 79 drawn with replacement.
First, recall a key interpretation: if you repeated this sampling and interval-construction process many times, about 85% of the resulting intervals would contain the true population mean. This is the defining property of a confidence interval with confidence level 0.85.
Next, consider the role of the sample mean x̄ in a single execution. For the given sample, x̄ = −1.6, so the interval produced would be −1.6 − 8.5 to −1.6 + 8.5, i.e., [−10.1, 6.9].
Whether this particular interval contains the true mean (which is 4.9 in Julie’s problem) is determined by the actual draw, but the long-run frequency claim remains tied to the confidence level: the method is designed so that 85% of such constructed intervals will surround the mean.
Why one option might be tempting but flawed: if someone tries to argue that the probability is entirely determined by the observed sample mean, that would ignore the random nature of the sampling process and the underlying construction of the interval; the interval’s success in this instance is a realized outcome within the 85% framework, not a guaranteed outcome for this single sample.
Also, note that the sample size (79) and the sampling with replacement are part of the method that ensures the sampling distribution used to justify the interval is appropriate; altering these assumptions would affect the coverage probability.
In summary, the essence is that the procedure is designed to have an 85% chance of capturing the true mean across repeated uses, and the specific observed x̄ determines the exact numeric interval for this trial, such as [−10.1, 6.9] for x̄ = −1.6.

BTRY3010/BTRY5010/STSCI2200/STSCI5200 Lecture#21 (Pages 102-103)

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