a-1: Population mean is unknown, only estimated through confidence intervals.
a-2: Best estimate is the sample mean of 35 kg.
b-1: We use the t-distribution because the population standard deviation is unknown.
b-2: Assume normal population and random sampling.
c: Value of t is 1.699 (rounded to 3 decimal places).
d: Confidence interval is 23.04 kg to 46.96 kg.
a-1. Population Mean:
- We can't directly estimate the population mean from a single sample. The best we can do is estimate its range with a confidence interval.
a-2. Best Estimate:
- The sample mean of 35 kg is the best estimate we have for the population mean, even though it's not a definitive value.
b-1. Using the t-distribution:
- We use the t-distribution instead of the normal distribution because we don't know the population standard deviation (only the sample standard deviation). The t-distribution is similar to the normal but has "fatter tails" to account for this uncertainty.
b-2. Assumptions:
- We need to assume that the population is normally distributed (although not perfectly) and that the sample is randomly selected from the population.
c. Value of t:
- With a 90% confidence interval and 27 degrees of freedom (n-1), the t-value is approximately 1.699 (rounded to 3 decimal places). You can find this value in a t-distribution table or online calculators.
d. Confidence Interval:
- The 90% confidence interval is:
35 kg ± (t * s / √n) ≈ 35 kg ± (1.699 * 13 kg / √27) ≈ 23.04 kg to 46.96 kg