Sample Of 16 Small Bags Of The Same Brand Of Candies

A Sample Of 16 Small Bags Of The Same Brand Of Candies Was Selected A

A sample of 16 small bags of the same brand of candies was selected. Assume that the population distribution of bag weights is normal. The weight of each bag was then recorded. The mean weight was 2 ounces with a standard deviation of 0.12 ounces. The population standard deviation is known to be 0.1 ounce. a.

Define the Random Variable X, in words. b. Which distribution should you use for this problem? Explain your choice. c. Construct a 90% confidence interval for the population average weight of the candies. d. State the confidence interval in words. e. Calculate the error bound (EBM). f. Construct a 98% confidence interval for the population average weight of the candies. g. State the confidence interval in words. h. Calculate the error bound (EBM). g. In complete sentences, explain why the confidence interval in (f) is larger than the confidence interval in (c).

Paper For Above instruction

The problem at hand involves estimating the average weight of candies in small bags using statistical inference. The data consists of a sample of 16 bags, with their weights recorded, showing a sample mean of 2 ounces and a known population standard deviation of 0.1 ounce. Such a scenario allows us to perform confidence interval calculations to estimate the true population mean based on sample data, assuming the population distribution is normal.

Part a: Definition of the Random Variable X

The random variable X can be defined as the weight of a randomly selected small bag of candies from the entire population of such bags. In other words, X represents the individual bag weight which follows a certain probability distribution, assumed to be normal based on the problem statement.

Part b: Appropriate Distribution

Given that the population standard deviation is known (σ = 0.1 ounce) and the population distribution of bag weights is normal, the appropriate distribution to use is the standard normal distribution (z-distribution). This is because when the population variance is known, and the distribution is normal, the z-test and confidence intervals are appropriate for analysis.

Part c: Constructing a 90% Confidence Interval

To construct a 90% confidence interval for the population mean, we use the formula:

CI = x̄ ± z* (σ/√n)

Where:

- x̄ = 2 ounces (sample mean)

- σ = 0.1 ounces (population standard deviation)

- n = 16 (sample size)

- z* = z-value corresponding to the 90% confidence level (from z-tables, approximately 1.645)

Calculating the standard error (SE):

SE = σ/√n = 0.1 / √16 = 0.1 / 4 = 0.025

The margin of error (EBM):

EBM = z* × SE = 1.645 × 0.025 ≈ 0.0411

Constructed confidence interval:

Lower limit: 2 - 0.0411 = 1.9589


Upper limit: 2 + 0.0411 = 2.0411

Thus, the 90% confidence interval is approximately (1.959, 2.041) ounces.

Part d: Interpretation of the 90% Confidence Interval

We are 90% confident that the true average weight of all bags of this brand of candies lies between approximately 1.959 ounces and 2.041 ounces.

Part e: Error Bound (EBM) for 90% Confidence Interval

The error bound, or margin of error, for the 90% confidence interval is approximately 0.0411 ounces.

Part f: Constructing a 98% Confidence Interval

Using the same formula:

- z* for 98% confidence is approximately 2.33 (from z-tables).

- Standard error remains 0.025.

- Margin of error:

EBM_98 = 2.33 × 0.025 ≈ 0.0583

Constructed 98% confidence interval:

Lower limit: 2 - 0.0583 ≈ 1.9417


Upper limit: 2 + 0.0583 ≈ 2.0583

Hence, the 98% confidence interval is approximately (1.942, 2.058) ounces.

Part g: Interpretation of the 98% Confidence Interval

We are 98% confident that the true average weight of all bags of this candy brand falls between approximately 1.942 ounces and 2.058 ounces.

Part h: Error Bound for 98% Confidence Interval

The error bound for the 98% confidence interval is approximately 0.0583 ounces.

Part i: Explanation of Why the 98% Interval is Larger Than the 90%

The confidence interval for the 98% confidence level is larger than that for the 90% level because increasing confidence requires capturing a broader range of possible true means. To achieve this higher confidence, the critical z-value increases from approximately 1.645 to 2.33, which proportionally expands the margin of error. Consequently, the interval widens, providing greater certainty that the true mean is within the interval but at the cost of precision. This trade-off between confidence level and interval width is a fundamental aspect of inferential statistics, illustrating how confidence levels influence the range of estimated population parameters.

References

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