A survey of college students was conducted during final exam week to assess the number of cups of coffee consumed each day. The mean number of cups was 5 with a standard deviation of 1.5 cups. The distribution was normal.
A. What proportion of students drank 7 or more cups of coffee per day?
B. What proportion of students drank 2 or more cups of coffee per day?
C. What proportion of students drank between 2 and 7 cups of coffee per day?
D. How many cups of coffee would an individual at the 60th percentile rank drink?
E. What is the percentile rank for an individual who drinks 4 cups of coffee a day?
F. What is the percentile rank for an individual who drinks 7.5 cups of coffee a day?
A, B, C, E, F. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
D, E, F. Percentile rank is proportion ≤ that score.
For D, start with the table to get the Z score, then use the above equation.
All it has in the back of my textbook is Appendix A where it states, "Areas Under the Normal Curve (Z-Table)." Is this the correct table and what exactly am I supposed to look for? There are no straight numbers because on this table is just numbers that are in decimals. The table doesn't have straight numbers such as 1, 2, 3, 4, 5, 6, etc.
A survey of college students was conducted during final exam week to assess the number of cups of coffee consumed each day. The mean number of cups was 5 with a standard deviation of 1.5 cups. The distribution was normal.
A. What proportion of students drank 7 or more cups of coffee per day?
B. What proportion of students drank 2 or more cups of coffee per day?
C. What proportion of students drank between 2 and 7 cups of coffee per day?
D. How many cups of coffee would an individual at the 60th percentile rank drink?
E. What is the percentile rank for an individual who drinks 4 cups of coffee a day?
F. What is the percentile rank for an individual who drinks 7.5 cups of coffee a day?
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To answer these questions, we will use the normal distribution properties and Z-scores. A Z-score measures how many standard deviations an observation is from the mean. We can use this information to find proportions and percentiles.
A. To find the proportion of students who drank 7 or more cups of coffee per day, we need to find the area under the normal curve to the right of 7 cups. First, we calculate the Z-score:
Z = (x - mean) / standard deviation
Z = (7 - 5) / 1.5
Z = 1.33
Next, we look up the area to the right of 1.33 in the standard normal distribution table or use a calculator. The area to the right of 1.33 is approximately 0.0918 or 9.18%. Therefore, approximately 9.18% of students drank 7 or more cups of coffee per day.
B. To find the proportion of students who drank 2 or more cups of coffee per day, we need to find the area under the normal curve to the right of 2 cups. Again, we calculate the Z-score:
Z = (x - mean) / standard deviation
Z = (2 - 5) / 1.5
Z = -2
Now, we look up the area to the right of -2 in the standard normal distribution table or use a calculator. The area to the right of -2 is approximately 0.9772. Since we are interested in the area to the left of -2, we subtract it from 1:
1 - 0.9772 = 0.0228 or 2.28%. Therefore, approximately 2.28% of students drank 2 or more cups of coffee per day.
C. To find the proportion of students who drank between 2 and 7 cups of coffee per day, we need to find the area between the Z-scores for 2 and 7. We calculate the Z-scores:
Z1 = (x1 - mean) / standard deviation = (2 - 5) / 1.5 = -2
Z2 = (x2 - mean) / standard deviation = (7 - 5) / 1.5 = 1.33
Next, we find the area to the right of -2 and the area to the right of 1.33, and subtract the smaller area from the larger area:
Area between Z1 and Z2 = Area(Z2) - Area(Z1)
= 0.0918 - 0.0228
= 0.069 or 6.9%. Therefore, approximately 6.9% of students drank between 2 and 7 cups of coffee per day.
D. To find the number of cups of coffee an individual at the 60th percentile rank would drink, we need to find the Z-score corresponding to the 60th percentile. We find this value using the standard normal distribution table or a calculator. The Z-score corresponding to the 60th percentile is approximately 0.253. We then solve for the number of cups using the formula:
Z = (x - mean) / standard deviation
0.253 = (x - 5) / 1.5
Solving for x gives:
0.253 * 1.5 + 5 = x
x ≈ 5.38
Therefore, an individual at the 60th percentile rank would drink approximately 5.38 cups of coffee per day.
E. To find the percentile rank for an individual who drinks 4 cups of coffee a day, we first calculate the Z-score:
Z = (x - mean) / standard deviation
Z = (4 - 5) / 1.5
Z = -0.67
We then find the area to the left of -0.67 in the standard normal distribution table or use a calculator. The area to the left of -0.67 is approximately 0.2514 or 25.14%. Therefore, an individual who drinks 4 cups of coffee a day is at the 25.14th percentile rank.
F. To find the percentile rank for an individual who drinks 7.5 cups of coffee a day, we calculate the Z-score:
Z = (x - mean) / standard deviation
Z = (7.5 - 5) / 1.5
Z = 1.33
Next, we find the area to the left of 1.33 in the standard normal distribution table or use a calculator. The area to the left of 1.33 is approximately 0.9088 or 90.88%. Therefore, an individual who drinks 7.5 cups of coffee a day is at the 90.88th percentile rank.