Question

The weight, in grams, of each of twelve randomly selected avocados are shown below. 153,161,149,155,158,144,152,148,145,152,159,147 Assume the weights of avocados are normally distributed. Find the estimate for the population mean weight of avocados in the form of a 90% confidence interval by addressing the five parts below.

A. Find the relevant sample statistics, mean and standard deviation. Express each answer as a decimal value rounded to the nearest hundredth (two digits to the right of the decimal point).

B. Find the value of the corresponding test statistic. Express the answer as a decimal value rounded to the nearest thousandth (three digits to the right of the decimal point).

C. Find the margin of error called the error bound for a population mean (EBM). Express the answer as a decimal value rounded to the nearest hundredth (two digits to the right of the decimal point).

D. Construct the confidence interval. Express the answer as a decimal value rounded to the nearest tenth (one digit to the right of the decimal point).

E. Write a sentence that interprets the estimate in the context of the problem.

Answer 1

To estimate the population mean weight of avocados with a 90% confidence interval, we determine the sample mean and standard deviation, find the critical t-value, calculate the margin of error, and construct the confidence interval, interpreting it to mean that there's a 90% confidence the true mean lies within the interval.

Step-by-step explanation:

To estimate the population mean weight of avocados with a 90% confidence interval we first need to calculate the sample mean and sample standard deviation. Then, we use the t-distribution since the population standard deviation is unknown.

Sample Mean and Standard Deviation

The sample mean (´X) is the sum of all sample weights divided by the number of avocados, which is:

´X = (153 + 161 + 149 + 155 + 158 + 144 + 152 + 148 + 145 + 152 + 159 + 147) / 12

To calculate the sample standard deviation (s), we use the formula for s = sqrt(Σ(xi - ´X)² / (n-1)), where xi is a sample value, and n is the sample size.

Test Statistic (t)

For a 90% confidence interval, we will need to look up the critical t-value corresponding to our confidence level and degrees of freedom (n-1).

Error Bound for Population Mean (EBM)

EBM = t * (s / sqrt(n)) which provides the margin of error for our confidence interval.

Confidence Interval

The confidence interval is calculated by taking the sample mean and adding and subtracting the EBM from it. This gives us the lower and upper bounds of the interval.

Interpretation

An interpretation of the confidence interval would be: "Based on the sample, we are 90% confident that the true population mean weight of avocados falls within the calculated interval."