A group of 894 women aged 70-79 had their height and weight measured. The mean height was 159 cm with a standard deviation of 5 cm and the mean weight was 65.9kg with a standard deviation of 12.7kg. Both sets of data are fairly normal.
A.) Suppose you were asked for a range of typical heights and weights for this population of women. What values would you give? Explain.
B.) Which of the two measurements appears more variable? Explain.
C.) What percentage of the population is expected to be taller than 166cm?
D.) What percentage of the population is expected to weigh between 55 and 75kg?
E.) Above what weight will 85% of the population lie?
A.) about 99% of a normally distributed population lies within 2½ standard deviations of the mean
159 cm ± (2½ * 5 cm)
65.9 kg ± (2½ * 12.7 kg)
B.) the weight seems more variable because the s.d. is a larger fraction of the mean
C.) 7 cm above the mean is 1.4 s.d.
this is about 8% of the population
D.) 55 is (10.9/12.7) s.d. below the mean, and 75 is (9.1/12.7) s.d. above
this is about 60% of the population
E.) this is slightly more than 1 s.d. below the mean
a weight of about 53 kg
This applet will be very useful
http://davidmlane.com/normal.html
You don't even have to find the z-scores, just enter the data as given
A.) To determine a range of typical heights and weights for this population of women, we can use the concept of standard deviations. Typically, data within one standard deviation of the mean is considered "typical" or "normal."
For the height, since the mean is 159 cm with a standard deviation of 5 cm, we can find the range of typical heights by adding and subtracting one standard deviation from the mean.
Typical height range = Mean height ± (1 * Standard deviation) = 159 cm ± (1 * 5 cm) = 154 cm to 164 cm.
Similarly, for the weight, since the mean is 65.9 kg with a standard deviation of 12.7 kg, the typical weight range would be:
Typical weight range = Mean weight ± (1 * Standard deviation) = 65.9 kg ± (1 * 12.7 kg) = 53.2 kg to 78.6 kg.
B.) To determine which measurement appears more variable, we can compare the standard deviations. The measurement with a larger standard deviation is considered more variable. In this case, the standard deviation for weight (12.7 kg) is larger than the standard deviation for height (5 cm). Thus, weight appears more variable in this population of women.
C.) To determine the percentage of the population expected to be taller than 166 cm, we can use z-scores and the normal distribution.
First, we need to calculate the z-score for a height of 166 cm:
z = (x - mean) / standard deviation
z = (166 - 159) / 5
z = 1.4
Next, we can consult the Z-table or use a calculator to find the percentage of the population taller than this z-score. From the Z-table, we find that the percentage is approximately 8.17%. Therefore, approximately 8.17% of the population is expected to be taller than 166 cm.
D.) To find the percentage of the population expected to weigh between 55 kg and 75 kg, we can again use z-scores and the normal distribution.
First, we need to calculate the z-scores for weights of 55 kg and 75 kg:
z1 = (55 - 65.9) / 12.7
z1 = -0.865
z2 = (75 - 65.9) / 12.7
z2 = 0.716
Next, we find the area under the curve between these two z-scores using the Z-table or calculator. From the Z-table, we find that the percentage is approximately 53.46%. Therefore, approximately 53.46% of the population is expected to weigh between 55 kg and 75 kg.
E.) To determine the weight above which 85% of the population lies, we can again use z-scores and the normal distribution.
First, we need to find the z-score corresponding to the 85th percentile. From the Z-table, we find that the z-score is approximately 1.036.
Then, we can use the z-score formula to find the corresponding weight:
z = (x - mean) / standard deviation
1.036 = (x - 65.9) / 12.7
Solving for x:
x - 65.9 = 1.036 * 12.7
x - 65.9 = 13.145
x = 65.9 + 13.145
x ≈ 79.045
Therefore, above a weight of approximately 79.045 kg, 85% of the population is expected to lie.
A.) To determine a range of typical heights and weights for this population of women, we can use the concept of standard deviation. The standard deviation provides a measure of how spread out the data is from the mean.
For height:
- We know that the mean height is 159 cm with a standard deviation of 5 cm.
- Typically, about 68% of the data falls within one standard deviation of the mean, and about 95% falls within two standard deviations.
- Based on this, we can expect that a range of typical heights for this population of women would be approximately 154-164 cm (mean ± one standard deviation).
For weight:
- We know that the mean weight is 65.9 kg with a standard deviation of 12.7 kg.
- Using the same logic as above, a range of typical weights for this population of women would be approximately 53.2-78.6 kg (mean ± one standard deviation).
B.) To determine which of the two measurements appears more variable, we can compare the standard deviations.
- For height, the standard deviation is 5 cm.
- For weight, the standard deviation is 12.7 kg.
- The weight measurement has a larger standard deviation, indicating greater variability in the data. Therefore, weight appears to be more variable within this population of women.
C.) To find the percentage of the population expected to be taller than 166 cm, we can use the concept of z-scores.
- First, we need to calculate the z-score for 166 cm using the formula: z = (x - mean) / standard deviation. In this case, x = 166 cm, mean = 159 cm, and standard deviation = 5 cm.
- Plugging in the values, we get z = (166 - 159) / 5 = 1.4.
- Using a standard normal distribution table or calculator, we can find the percentage associated with a z-score of 1.4. This value represents the percentage of the population expected to be taller than 166 cm.
D.) To find the percentage of the population expected to weigh between 55 and 75 kg, we can again use z-scores.
- First, we need to calculate the z-scores for 55 kg and 75 kg using the formula: z = (x - mean) / standard deviation. In this case, x = 55 kg and 75 kg, mean = 65.9 kg, and standard deviation = 12.7 kg.
- Plugging in the values, we find z1 = (55 - 65.9) / 12.7 and z2 = (75 - 65.9) / 12.7.
- Using a standard normal distribution table or calculator, we can find the percentage associated with each z-score. The difference between these two percentages represents the percentage of the population expected to weigh between 55 and 75 kg.
E.) To determine the weight above which 85% of the population will lie, we can again use z-scores.
- We want to find the z-score associated with the 85th percentile, which is the value below which 85% of the data lies.
- Using a standard normal distribution table or calculator, we can find the z-score that corresponds to the 85th percentile.
- Once we have the z-score, we can use the formula z = (x - mean) / standard deviation to calculate the weight value, where x is the weight we're looking for, mean is the mean weight of 65.9 kg, and standard deviation is 12.7 kg.