Final answer:
The appropriate 90% confidence interval for the true mean life of heat pumps, using a sample size of 6 and a sample standard deviation of 1.93, is found to be approximately (1.795, 4.971), which corresponds to Option D.
Step-by-step explanation:
To find the confidence interval for the mean life of heat pumps given a sample with a sample standard deviation of 1.93, a sample size of 6, and using a significance level of 0.10 (which corresponds to a 90% confidence level), we can use the t-distribution since the sample size is small. First, we calculate the sample mean:
Mean = (2.0 + 1.3 + 6.0 + 1.9 + 5.1 + 4.0) / 6 = 20.3 / 6 = 3.3833
Next, we find the t-value for a 90% confidence interval and degrees of freedom (n - 1), which is 5. From the t-distribution table or using a calculator with statistical functions, we obtain the t-value (we round to two decimal places for simplicity here).
For a 90% confidence interval and 5 degrees of freedom, the t-value is approximately 2.015.
Now, calculate the margin of error (E):
E = (t-value) × (s / sqrt(n))
E = 2.015 × (1.93 / sqrt(6)) = 2.015 × (1.93 / 2.4495) = 1.58 (approximately)
Finally, we can find the confidence interval:
Lower limit = Mean - E = 3.3833 - 1.58 = 1.8033
Upper limit = Mean + E = 3.3833 + 1.58 = 4.9633
Thus, we can round our answer to two decimal places and select the correct option:
The appropriate confidence interval for the true mean life of heat pumps is approximately (1.80, 4.96), which corresponds to Option D. (1.795, 4.971).