A manufacturer wishes to estimate the reliability (in months) of a certain product. An estimate of the population standard deviation from a previous sample was 12 months. If the manufacturer desires to be within 4 months of the true value with approximately 80% confidence, what should the sample size be?

To determine the sample size needed to estimate the reliability of the product with the desired level of confidence and precision, we can use the formula for sample size estimation in a confidence interval:

n = (Z * σ / E)^2

Where:
n = sample size
Z = Z-value (also known as the z-score) corresponding to the desired level of confidence
σ = population standard deviation
E = margin of error (desired precision)

In this case, the desired level of confidence is approximately 80%, which corresponds to a z-score of 1.28 (assuming a normal distribution). The margin of error (E) is given as 4 months, and the population standard deviation (σ) is 12 months.

Substituting these values into the formula, we get:

n = (1.28 * 12 / 4)^2
n = (15.36 / 4)^2
n ≈ 3.84^2
n ≈ 14.75

Therefore, the sample size needed to estimate the reliability of the product with an approximately 80% confidence and a margin of error of 4 months is approximately 15.

To determine the sample size required to estimate the reliability of the product, we can use the formula for sample size for estimating means:

n = ((Z * σ) / E)^2

Where:
n = sample size
Z = Z-value for the desired confidence level (80% confidence corresponds to a Z-value of approximately 1.28)
σ = population standard deviation
E = the maximum error tolerance (half the desired confidence interval width)

In this case, the desired maximum error tolerance (E) is 4 months, and the population standard deviation (σ) is given as 12 months. Let's plug these values into the formula:

n = ((1.28 * 12) / 4)^2
n = (15.36 / 4)^2
n ≈ 3.84^2
n ≈ 14.75

Thus, the sample size required to estimate the reliability of the product with approximately 80% confidence and a maximum error tolerance of 4 months is approximately 15 (rounded up).

Note: Since sample sizes must be whole numbers, we rounded up to the nearest whole number in this case.

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