Question

A supermarket chain packages ground beef using meat trays of two sizes: one that holds approximately 1 pound of meat, and one that holds approximately 3 pounds. A random sample of 37 packages in the smaller meat trays produced weight measurements with an average of 1.01 pounds and a standard deviation of 0.12 pound.

(a) Construct a 99% confidence interval for the average weight of all packages sold in the smaller meat trays by this supermarket chain. (Round your answers to three decimal places.)

___ Ib to ___lb

(b) What does the phrase "99% confident" mean? There is a 99% chance that an individual sample mean will fall within the interval. In repeated sampling, 1% of all intervals constructed in this manner will enclose the population mean, In repeated sampling, 99% of all intervals constructed in this manner will enclose the population mean. 99% of all values will fall within the interval. There is a 1% chance that an individual sample mean will fall within the interval.

(c) Suppose that the quality control department of this supermarket chain intends that the amount of ground beef in the smaller trays should be 1 pound on average. Should the confidence interval in part (a) concern the quality control department? Explain.

Based on the confidence interval, the quality control department -Select be concerned. Since 1 is -Select-- the interval, the interval - Select that is different from 1.

Answer 1

a) 99% Confidence interval = (0.956, 1.064) in pounds.

99% confidence interval for the average weight of all packages sold in the smaller meat trays by this supermarket chain.

0.956 Ib to 1.064 lb.

b) The meaning of 99% confidence level is that: In repeated sampling, 99% of all intervals constructed in this manner will enclose the population mean.

c) No, the confidence interval obtained in (a) shouldn't really concern the quality control department.

Based on the confidence interval, the quality control department should not be concerned. Since 1 is properly within the interval, the interval doesn't show that the population mean can be (very) different from 1.

Explanation:

a) Confidence Interval for the population mean is basically an interval of range of values where the true population mean can be found with a certain level of confidence.

Mathematically,

Confidence Interval = (Sample mean) ± (Margin of error)

Sample Mean = 1.01 lb

Margin of Error is the width of the confidence interval about the mean.

It is given mathematically as,

Margin of Error = (Critical value) × (standard Error of the mean)

Critical value will be obtained using the t-distribution. This is because there is no information provided for the population mean and standard deviation.

To find the critical value from the t-tables, we first find the degree of freedom and the significance level.

Degree of freedom = df = n - 1 = 37 - 1 = 36.

Significance level for 99% confidence interval

(100% - 99%)/2 = 0.5% = 0.005

t (0.005, 36) = 2.719 (from the t-tables)

Standard error of the mean = σₓ = (σ/√n)

σ = standard deviation of the sample = 0.12 lb

n = sample size = 37

σₓ = (0.12/√37) = 0.0197

99% Confidence Interval = (Sample mean) ± [(Critical value) × (standard Error of the mean)]

CI = 1.01 ± (2.719 × 0.0197)

CI = 1.01 ± 0.0536

99% CI = (0.9564, 1.0636)

99% Confidence interval = (0.956, 1.064)

b) Just like I explained at the beginning of part (a), the meaning of 99% confidence level is that: In repeated sampling, 99% of all intervals constructed in this manner will enclose the population mean. This is the option that is the most correct among the options provided.

c) Based on the confidence interval obtained in (a), all the population averages for weight of meat (ground beef) will vary between 0.956 lb and 1.064 lb. The required average by the quality control department (1 lb) falls properly within this interval, so, there shouldn't be much cause for concern.

Hope this Helps!!!