An apple grower is trying to produce larger fruit by using a new fertilizer. The usual fertilizer produces fruit with a mean diameter of 8.7 cm. Two new fertilizer brands are under consideration, with the following experimental results:
BRAND A: n = 10, mean = 9.5 cm, s = 4 cm BRAND B: n = 10, mean = 10.8 cm, s = 3 cm
Calculate the 98% confidence interval for the true difference between the means of the two brands. [5]
Do you think that Brand B is significantly better than Brand A?
To calculate the 98% confidence interval for the true difference between the means of the two brands, we can use the two-sample t-test formula:
Confidence interval = (Mean of Brand B - Mean of Brand A) ± (t-critical value * Standard error of the difference)
1. Calculate the standard error of the difference:
Standard error of the difference = sqrt((s1^2 / n1) + (s2^2 / n2))
For BRAND A: s1 = 4 cm and n1 = 10
For BRAND B: s2 = 3 cm and n2 = 10
Standard error of the difference = sqrt((4^2 / 10) + (3^2 / 10))
2. Find the t-critical value for a 98% confidence interval with (n1 + n2 - 2) degrees of freedom.
Since n1 = 10 and n2 = 10, degrees of freedom = (10 + 10 - 2) = 18
You can consult a t-distribution table or use a statistical software to find the t-critical value for a 98% confidence level with 18 degrees of freedom.
3. Calculate the confidence interval:
Mean of Brand A = 9.5 cm
Mean of Brand B = 10.8 cm
Confidence interval = (10.8 - 9.5) ± (t-critical value * standard error of the difference)
Finally, calculate the range for the confidence interval and determine if it includes zero.
To determine if Brand B is significantly better than Brand A, check if the confidence interval includes zero. If the confidence interval does not include zero, it suggests that there is a significant difference between the means of the two brands.
Remember, you will need specific statistical information (t-distribution table or software) to find the t-critical value and perform the calculations.