A survey was taken of Australian companies that do business with companies in China. One of the survey questions was ‘Approximately how many years has your company been trading with companies in China?’

A random sample of 44 responses to this question yielded a mean of 10.455 years. Suppose that the population standard deviation is 7.7 years. For the population of Australian companies that do business with Chinese companies, what is the 90% Confidence Interval of the mean number of years that a company has been trading with China?

Calculate the Lower limit

Question

A survey was taken of Australian companies that do business with companies in China. One of the survey questions was ‘Approximately how many years has your company been trading with companies in China?’

A random sample of 44 responses to this question yielded a mean of 10.455 years. Suppose that the population standard deviation is 7.7 years. For the population of Australian companies that do business with Chinese companies, what is the 90% Confidence Interval of the mean number of years that a company has been trading with China?

Calculate the Lower  limit

BlackTom AI · Accepted Answer

We are asked for the 90% confidence interval for the population mean number of years Australian companies have been trading with China, given a known population standard deviation.
First, identify the given values: sample size n = 44, sample mean x̄ = 10.455, population standard deviation σ = 7.7. Since σ is known and the confidence level is 90%, we use the z-based formula for the confidence interval: x̄ ± z_{α/2} * (σ / sqrt(n)).
Next, determine the critical z-value for a 90% CI. A 90% central interval leaves 5% in each tail, so z_{0.95} ≈ 1.645. This is the factor we'll multiply by the standard error.
Compute the standard error: SE = σ / sqrt(n) = 7.7 / sqrt(44). Calculate sqrt(44) ≈ 6.6332496, so SE ≈ 7.7 / 6.6332496 ≈ 1.160.
Now multiply by the z-score to get the margin of error: MOE = z * SE ≈ 1.645 * 1.160 ≈ 1.909.
Finally, construct the lower limit: Lower = x̄ − MOE ≈ 10.455 − 1.909 ≈ 8.546. When rounding to four decimals, this is 8.5455 (depending on rounding during intermediate steps).
So the 90% confidence interval for the mean years is centered at 10.455 with a margin of about 1.909, giving a lower limit near 8.546. The key steps were selecting the correct z for 90% (1.645), calculating SE with the known σ and n, and subtracting the margin from the observed mean to obtain the lower bound.

BSTA011 Week 7 Practice Quiz

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