Question

1. An article in Food Testing and Analysis "Improving Reproducability of Refractometry Measurements of Fruit Juices" (1999, Vol. 4, No. 4, pp. 13-17) measured the sugar concentration (Brix) in clear apple juice. All measurements were taken at 20°C: 11.48, 11.45, 11.48, 11.47, 11.48, 11.50, 11.42, 11.49 Test the hypothesis H:u= 11.5 versus H :u # 11.5 using a = .05. (a) Reject H, since -3.21 < -1.8946 (b) Reject Ho since - 3.21 < -1.96 (c) Reject H, since -3.86 < -2.365 (d) Reject H, since -3.21 < -2.365 (e) Reject Hsince - 3.86 < -1.96 2. (Continuation of 1.) Based on your conclusion in 1., which of the following is true? (a) A Type I error might ahve occurred (b) A Type II error might have occurred (c) The population mean sugar concentration is not equal to 11.5 (d) The probability of Type I error is .025. (e) The probability of Type II error is .95 3. (Continuation of 1.) Find the p-value. (a).005< P<.01 (b).01 <P < .025 (e).01 < P < .02 (d).0007 (e).0014

Please provide step by step solution for these questions. Thank you.

Answer 1

Soln

1)

Option d ie Reject H₀ since -3.21 < -2.365

From the data, we calculate the mean and standard deviation.

Mean (X bar) = Sum of Values /n

and

Σ'(x-x) Ο = 1 η - 1

Hence,

Mean (x) = 11.47

Std Dev (s) = 0.025

n = 8

alpha = 0.05

Using the Critical t table,

t_Critical = -2.365 or 2.365

Null and Alternate Hypothesis

H₀: µ = 11.5

H_a: µ <> 11.5

Test Statistic

t = (x - µ₀) / (s/n^1/2) = (11.47 -11.5) / (0.025/8^1/2) =-3.21

p-value = TDIST(3.21,8-1,2) = 0.0148

Result

Since the p-value is less than 0.05 (and t value is less than -2.365) , we reject the null hypothesis.

2)

Option A ie type I error

Type I error is the rejection of a true null hypothesis. We cannot calculate the probability of Type I Error as the true mean is not provided.

3)

Option C

p-value = TDIST(3.21,8-1,2) = 0.0148