Many food products contain small quantities of substances that would give an und
ID: 3174063 • Letter: M
Question
Many food products contain small quantities of substances that would give an undesirable 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 enology 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 enology.
20 30 35 26 23 31 34 43 36 32
Assume (this is not realistic) that the standard deviation of the odor threshold for untrained noses is known to be = 7 µg/l. A normal quantile plot confirms that there are no systematic departures from normality.
(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 enology 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 = 0.05. Round your value for z to two decimal places and round your P-value to four decimal places.)
z =
P-value =
Explanation / Answer
Result:
Many food products contain small quantities of substances that would give an undesirable 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 enology 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 enology.
20 30 35 26 23 31 34 43 36 32
Assume (this is not realistic) that the standard deviation of the odor threshold for untrained noses is known to be = 7 µg/l. A normal quantile plot confirms that there are no systematic departures from normality.
(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
Stem and Leaf plot for
Data
stem unit =
10
leaf unit =
1
Frequency
Stem
Leaf
3
2
0 3 6
6
3
0 1 2 4 5 6
1
4
3
10
Plot shows that the distribution is roughly symmetric with no outliers.
(b) Give a 95% confidence interval for the mean DMS odor threshold among all beginning enology students. (Round your answers to three decimal places.) ,
Mean =310/10=31
Confidence Interval Estimate for the Mean
Data
Population Standard Deviation
7
Sample Mean
31
Sample Size
10
Confidence Level
95%
Intermediate Calculations
Standard Error of the Mean
2.2136
Z Value
1.9600
Interval Half Width
4.3386
Confidence Interval
Interval Lower Limit
26.661
Interval Upper Limit
35.339
(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 = 0.05. Round your value for z to two decimal places and round your P-value to four decimal places.)
z =2.71
P-value = 0.0034
Z Test of Hypothesis for the Mean
Data
Null Hypothesis m=
25
Level of Significance
0.05
Population Standard Deviation
7
Sample Size
10
Sample Mean
31
Intermediate Calculations
Standard Error of the Mean
2.2136
Z Test Statistic
2.7105
Upper-Tail Test
Upper Critical Value
1.645
p-Value
0.0034
Reject the null hypothesis
Calculated P=0.0034 < 0.05 level of significance.
Ho is rejected.
We conclude that the mean odor threshold for beginning students is higher than the published threshold, 25 µg/l.
Stem and Leaf plot for
Data
stem unit =
10
leaf unit =
1
Frequency
Stem
Leaf
3
2
0 3 6
6
3
0 1 2 4 5 6
1
4
3
10
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