Question

In order to evaluate the effectiveness of a new type of plant food that was developed for tomatoes, a study was conducted in which a random sample of n = 68 plants received a certain amount of this new type of plant food each week for 14 weeks. The variable of interest is the number of tomatoes produced by each plant in the sample. The table below reports the descriptive statistics for this study:

Descriptive statistics for the tomato food study

Variable	n	sample mean	sample standard deviation	standard error
number of tomatoes	68	60.00	9.60	1.164

Assuming the population is normally distributed, the investigators would like to construct a 98% confidence interval for the average number of tomatoes that all plants of this variety can produce when fed this supplement like this.
a) The margin of error is:  (3 decimals)

b) The corresponding 98% confidence interval for the true population mean is:
Lower Limit:  (3 decimals)
to
Upper Limit:  (3 decimals)
c) What would we conclude at α = 0.02 for the hypothesis test H₀: μ = 64.875 vs. H_a: μ ≠ 64.875?

We do not have enough evidence to conclude the true mean is 64.875.We have insufficient evidence to conclude the true mean is different from 64.875.    We do not have enough evidence to conclude the true mean is 60.00.We have enough evidence to conclude that the true mean is 64.875.We have sufficient evidence to conclude that the true mean is different from 64.875.

Answer 1

Given
X̅ = 60           ....... Sample Mean
n = 68           ....... Sample Size
s = 9.60           ....... Sample Standard Deviation

SE = standard error = 1.164

Since the population standard deviation is unknown, we use the t-distribution

Degrees of Freedom = df = n - 1 = 68 - 1 = 67

For 98% Confidence interval

α = 0.02,      α/2 = 0.01
From t tables of Excel function T.INV.2T (α, degrees of freedom) we find the t value
t = T.INV.2T (0.02, 67) = 2.383
We take the positive value of t

a) Margin of Error = ME is given by

b)

Confidence interval is given by

= (57.226, 62.774)

c)

H0: μ = 64.875

Ha: μ ≠ 64.875

64.875 > 62.774,

that is 64.875 is greater than the Upper Limit of the 98% confidence interval

Thus, the confidence interval does not include 64.875

Hence, we reject the null hypothesis Ho

Answer : (Last option)

We have sufficient evidence to conclude that the true mean is different from 64.875.