Question

Many food products contain small quantities of substances that would give an undesireable taste or smell if they are present in large amounts. An example is the "off-odors" caused by sulfur compounds in wine. Oenologists (wine experts) have determined the odor threshold, the lowest concentration of a compound that the human nose can detect. For example, the odor threshold for dimethyl sulfide (DMS) is given in the oenology literature as 25 micrograms per liter of wine (µg/l). Untrained noses may be less sensitive, however. Here are the DMS odor thresholds for 10 beginning students of oenology. 33 26 32 23 35 22 31 41 36 30 Assume (this is not realistic) that the standard deviation of the odor threshold for untrained noses is known to be σ = 7 µg/l. (a) Make a stemplot to verify that the distribution is roughly symmetric with no outliers. (A normal quantile plot confirms that there are no systematic departures from normality. Enter numbers from smallest to largest, separated by spaces. Enter NONE for stems with no values.) 2 2 3 3 4

(b) Give a 95% confidence interval for the mean DMS odor threshold among all beginning oenology students. (Round your answers to three decimal places.)

________, _________

(c) Are you convinced that the mean odor threshold for beginning students is higher than the published threshold, 25 µg/l? Carry out a significance test to justify your answer. (Use α = 0.05. Round your value for z to two decimal places and round your P-value to four decimal places.)

z =

P-value =

State your conclusion. Reject the null hypothesis. There is significant evidence that the mean odor threshold for beginning students is higher than the published threshold. Reject the null hypothesis. There is not significant evidence that the mean odor threshold for beginning students is higher than the published threshold. Fail to reject the null hypothesis. There is significant evidence that the mean odor threshold for beginning students is higher than the published threshold. Fail to reject the null hypothesis. There is not significant evidence that the mean odor threshold for beginning students is higher than the published threshold.

Answer 1

The statistical software output for this problem is:

One sample Z hypothesis test:

μ : Mean of variable
H₀ : μ = 25
H_A : μ > 25
Standard deviation = 7

Hypothesis test results:

Variable	n	Sample Mean	Std. Err.	Z-Stat	P-value
Data	10	30.9	2.2135944	2.6653483	0.0038

Variable	n	Sample Mean	Std. Err.	L. Limit	U. Limit
Data	10	30.9	2.2135944	26.561435	35.238565

Hence,

a) Stem and Leaf plot:

2 : 2 3
2 : 6
3 : 0 1 2 3
3 : 5 6
4 : 1

b) 95% confidence interval:

(26.561, 35.239)

c) z = 2.67

P - value = 0.0038

d) Reject the null hypothesis. There is significant evidence that the mean odor threshold for beginning students is higher than the published threshold. Option A is correct.