A study of peach trees found that the average number of peaches per tree was 625. The standard deviation of the population is 140 peaches per tree. Ascientist wishes to find 80% confidence interval for the mean number of peaches per tree. How many trees does she need to sample to obtain an average accurate to within 16 peaches per tree?
Jim Tree sells Christmas trees. The mean length of the trees purchased was 68 inches with a standard deviation of 10 inches. Jim wants to know what per cent of his sales were more than 84 inches tall. He can use the standard normal distribution to help him.
To find out how many trees the scientist needs to sample to obtain an average accurate to within 16 peaches per tree, we can use the formula for determining sample size in a confidence interval.
The formula for sample size in a confidence interval is given by:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score associated with the desired level of confidence (80% confidence corresponds to a Z-score of 1.28)
σ = standard deviation of the population
E = desired margin of error (16 peaches per tree)
Plugging in the values, we have:
n = (1.28 * 140 / 16)^2
Simplifying the calculation:
n = (179.2 / 16)^2
n = 11.2^2
n ≈ 125.44
Since we cannot have a fraction of a tree, we round up the result to the nearest whole number. Therefore, the scientist needs to sample approximately 126 trees to obtain an average accurate to within 16 peaches per tree with 80% confidence.