What price do farmers get for the peach crops? in the third week of June, a random sample of 40 farming regions gave a sample mean of $6.88 per basket. assume that the standard deviation is known to be $1.92 per basket. find a 90% confidence interval for the population mean price per basket that farmers in this region get for their peach crop.
6.88
To find the confidence interval for the population mean price per basket, we can use the sample mean, sample size, and known standard deviation.
Here are the steps to calculate the confidence interval:
Step 1: Identify the sample mean, sample size, and standard deviation.
Given:
Sample mean (x-bar) = $6.88 per basket
Sample size (n) = 40
Standard deviation (σ) = $1.92 per basket
Step 2: Determine the confidence level.
The confidence level is given to be 90%, which means we can use a z-score table to find the critical value.
Step 3: Find the critical value (z-score).
Since the sample size is large (n > 30) and we know the population standard deviation, we can use the standard normal distribution (z-distribution) to find the critical value.
For a 90% confidence level, the critical value corresponds to a 5% (1 - 0.90) significance level in each tail. Looking up the z-score table, the critical value is approximately 1.645.
Step 4: Calculate the margin of error.
The margin of error (E) is calculated using the formula:
E = Critical Value * (Standard Deviation / √(Sample Size))
In this case:
E = 1.645 * ($1.92 / √(40))
Step 5: Calculate the confidence interval.
The confidence interval formula is:
Confidence Interval = Sample Mean ± Margin of Error
In this case:
Confidence Interval = $6.88 ± E
Now we can substitute the values and calculate the confidence interval.
E = 1.645 * ($1.92 / √(40)) ≈ 0.6609
Confidence Interval = $6.88 ± 0.6609
Therefore, the 90% confidence interval for the population mean price per basket that farmers in this region get for their peach crop is approximately ($6.22, $7.54).