Questions: The amount of time that it takes for various slow-growing tumors to double in size are listed in the table on the right. a. Find the mean and standard deviation of these data. b. How many of these cancers have doubling times that are within 2 standard deviations of the mean? c. If a person had a nonspecified tumor that was doubling every 230 days, discuss whether this particular tumor is growing at a rate that would be expected. Type of Cancer Doubling Time (days) Breast cancer 85 Rectal cancer 95 Synovioma 123 Skin cancer 133 Lip cancer 144 Testicular cancer 156 Esophageal cancer 165

Solution Steps

The mean \( \mu \) of the doubling times is calculated as follows:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{901}{7} = 128.7 \]

Thus, the mean of the doubling times is \( 128.7 \) days.

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = 770.5 \]

The standard deviation \( \sigma \) is then:

\[ \sigma = \sqrt{770.5} = 27.8 \]

Therefore, the standard deviation of the doubling times is \( 27.8 \) days.

To find how many cancers have doubling times within 2 standard deviations of the mean, we calculate the bounds:

\[ \text{Lower bound} = \mu - 2\sigma = 128.7 - 2 \times 27.8 = 73.1 \] \[ \text{Upper bound} = \mu + 2\sigma = 128.7 + 2 \times 27.8 = 184.3 \]

All 7 cancers fall within this range, so the number of cancers within 2 standard deviations is \( 7 \).

The Z-score \( z \) for a tumor doubling every 230 days is calculated as follows:

\[ z = \frac{X - \mu}{\sigma} = \frac{230 - 128.7}{27.8} = 3.6439 \]

Since the Z-score \( 3.6439 \) is greater than \( 2 \), we conclude that:

A tumor doubling every 230 days is not within the expected range (beyond 2 standard deviations).

Final Answer