The TreeDropped Fruit Company wants to sell its apples in pink and yellow gift boxes. To design the boxes, the company needs to estimate the average diameter of their apples. A random sample of 50 apples has a mean of 4.1 inches. Using previous research you assume a population standard deviation of .4 inches.
What is the point estimate in this scenario? Is it an unbiased estimator?
What is the critical z value for a 95% interval?
What is the margin of error for a 95% interval in this scenario?
What is the 95% confidence interval for this scenario?
I assume you are only concerned with the apples being too big to fit in the box.
Ho: mean = 4.1
Ha: mean > 4.1
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.05) and its Z score.
95% = mean + Z SEm
SEm = SD/√n
To find the answers to these questions, we can use statistics and confidence intervals. Let's go step by step:
1. Point Estimate:
The point estimate in this scenario refers to the mean diameter of the apples, which is found in the sample. The given sample mean is 4.1 inches, so the point estimate is 4.1 inches.
2. Unbiased Estimator:
For an estimator to be unbiased, the expected value of the estimator should equal the true population parameter. In this case, since we assume the population standard deviation is .4 inches and it is given that the sample is randomly selected, the sample mean is an unbiased estimator of the population mean.
3. Critical Z value for a 95% interval:
To find the critical Z value for a 95% confidence interval, we can use a standard normal distribution table or a statistical calculator. The Z value associated with a 95% confidence interval is approximately 1.96.
4. Margin of error for a 95% interval:
The margin of error represents the maximum likely difference between the sample estimate and the true population parameter. It can be calculated by multiplying the critical Z value by the standard deviation of the sample mean. In this scenario, the population standard deviation is given as 0.4 inches, so the margin of error would be 1.96 * (0.4 / √50). Simplifying this equation gives us the margin of error.
5. 95% Confidence interval:
The 95% confidence interval can be calculated by adding and subtracting the margin of error from the point estimate. In this scenario, the margin of error can be used to find the lower and upper bounds of the interval. The 95% confidence interval estimate for the population mean would be the point estimate plus/minus the margin of error.
By applying the formulas and calculations mentioned above, you should be able to find the necessary values for this scenario.