Construct A 90% Confidence Interval For The Mean Shell Stren

Construct a 90 confidence interval for the mean shell strength

Construct a 90% confidence interval for the mean shell strength

SmithCo is a supplier to many brand-name makers of mobile phones. SmithCo manufactures the external shells that enclose mobile phones. The strength of a shell is measured by applying increasing pressure to the shell and recording the pressure at which the shell breaks. Shell strengths are approximately normally distributed. A random sample of 10 shells made by SmithCo showed an average strength of 45.0 pounds and a standard deviation of 2.0 pounds. Use StatTools, or conduct calculations by hand, to construct a 90% confidence interval for the mean shell strength. The appropriate multiplier is 1.833.

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Constructing a confidence interval for the mean shell strength involves statistical reasoning based on sample data. Given the sample mean of 45.0 pounds, a sample standard deviation of 2.0 pounds, and a sample size of 10 shells, the goal is to estimate the range within which the true mean shell strength lies with 90% confidence.

Since the sample size is small (n

The confidence interval is calculated by taking the sample mean and adding and subtracting the product of the standard error and the multiplier. The standard error (SE) is computed as:

SE = s / sqrt(n)

where s is the sample standard deviation and n is the sample size. Plugging in the values:

SE = 2.0 / sqrt(10) ≈ 2.0 / 3.1623 ≈ 0.6325

The margin of error (ME) is then:

ME = multiplier × SE = 1.833 × 0.6325 ≈ 1.159

Therefore, the confidence interval in the form a ± b is:

45.0 ± 1.159

which means the lower bound is approximately 43.841 pounds, and the upper bound is approximately 46.159 pounds.

Alternatively, in the form [c, d], the interval is:

[43.841, 46.159]

This interval provides a range within which we are 90% confident the true mean shell strength of all shells produced by SmithCo falls, based on this sample data.

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