Question

The Biology Data Book reports that the gestation time for human babies averages 278 days with a standard deviation of 12 days. \(^{8}\) Suppose that these gestation times are normally distributed. a. Find the upper and lower quartiles for the gestation times. b. Would it be unusual to deliver a baby after only 6 months of gestation? Explain.

Short answer

Answer: The lower quartile of gestation time for human babies is approximately 269.9 days, and the upper quartile is approximately 286.1 days. Yes, it would be unusual for a baby to be delivered after only 6 months of gestation (approximately 182.64 days) as it is highly improbable based on the z-score of -7.96.

Step-by-step solution

Find the z-scores for the upper and lower quartiles

To find the z-scores for the upper and lower quartiles, we need to find the values corresponding to 25% (for the lower quartile) and 75% (for the upper quartile) of the area under the standard normal distribution curve. We can look up these values in a standard normal distribution table or use a calculator with a z-score function. Lower quartile z-score (25th percentile): \(z_{L} = -0.674\) Upper quartile z-score (75th percentile): \(z_{U} = 0.674\)

Find the gestation times for the upper and lower quartiles

Now, we can use these z-scores together with the mean and standard deviation of the gestation times to find the actual gestation times for the upper and lower quartiles. Gestation time for the lower quartile: \(Q_{1} = \mu + z_{L} * \sigma = 278 + (-0.674) * 12 = 278 - 8.088 = 269.912\) Gestation time for the upper quartile: \(Q_{2} = \mu + z_{U} * \sigma = 278 + 0.674 * 12 = 278 + 8.088 = 286.088\) So, the lower quartile is approximately 269.9 days and the upper quartile is approximately 286.1 days.

Find the probability of delivering a baby after only 6 months of gestation

First, let's convert 6 months of gestation to days. There are 30.44 days in a month, so 6 months approximately equals 6 * 30.44 = 182.64 days. Now, we need to find the z-score for 182.64 days: \(z = \frac{x - \mu}{\sigma} = \frac{182.64 - 278}{12} = -7.96\) The z-score of -7.96 is extremely far from the mean, which indicates that delivering a baby after only 6 months of gestation is highly improbable. So yes, it would be unusual to deliver a baby after only 6 months of gestation.

Key concepts

Understanding Z-scores

The Z-score is a statistical measurement that conveys how many standard deviations a specific data point is from the overall mean. In simpler terms, it's a way to express where a data point stands in relation to a dataset. For example, if your Z-score is 0, the data point is exactly at the mean. A positive Z-score indicates a value above the mean, while a negative Z-score shows a value below the mean. This is crucial when working with normally distributed data, as it allows you to understand the relative standing of data points.

In the case of gestation times, we used Z-scores to determine the position of the lower and upper quartiles within the distribution. The standard Z-scores for quartiles are often found in tables or calculated using statistical tools:

• Lower quartile (25th percentile): Z-score = -0.674
• Upper quartile (75th percentile): Z-score = 0.674

By using these, we can estimate corresponding actual gestation times, accurately reflecting the spread of the dataset.

Exploring Quartiles

Quartiles are an integral part of descriptive statistics that help in understanding how data is distributed in a dataset. They divide the dataset into four equal parts. Each part represents a quarter of the data. When considering normally distributed data, quartiles are particularly helpful in giving insights into the proportion of data that lies below or above certain values.

- **Lower Quartile (Q1):** This is the median of the first half of the dataset. It marks a value below which 25% of the data lies. In the gestation time example, Q1 was calculated as 269.912 days.
- **Upper Quartile (Q3):** This is the median of the second half of the dataset, providing a value below which 75% of the data can be found. We found Q3 for gestation to be approximately 286.088 days.
Understanding quartiles allows for a deeper look into data variability and patterns without analyzing each individual data point.

The Role of Standard Deviation

Standard deviation is a measure that indicates the amount of variation or dispersion in a set of values. The larger the standard deviation, the more spread out the values are around the mean. In the context of a normal distribution, a smaller standard deviation would mean the data is closely clustered around the mean.

In our example of gestation periods, the standard deviation is 12 days. Using this value:

• We can calculate how far gestation times deviate from the average of 278 days.
• It helps in determining Z-scores, as it's a key component in the formula: \(z = \frac{x - \mu}{\sigma}\).

A consistent standard deviation establishes reliability when predicting data behavior around the mean. In practice, this means predicting whether certain outcomes, like early deliveries, are outliers based on how rarely they occur in the distribution pattern.