Question

the lengths of pregnancy terms for a particular species of mammal are nearly normally distributed about a mean pregnancy length with a standard deviation of 9 days. about what percentage of births would be expected to occur within 9 days of the mean pregnancy length?

about % of births would be expected to occur within 9 days of the mean pregnancy length.
(type an integer or a decimal. round to two decimal places as needed.)

Explanation:

Identify the given parameters

We are given that the lengths of pregnancy terms for a particular species of mammal are nearly normally distributed.

• The standard deviation of the pregnancy length is \(\sigma = 9\) days.
• We want to find the percentage of births expected to occur within \(9\) days of the mean pregnancy length \(\mu\).

Relate the interval to standard deviations

The question asks for the percentage of births within \(9\) days of the mean \(\mu\).
Since the standard deviation \(\sigma\) is exactly \(9\) days, the interval "within \(9\) days of the mean" corresponds to:

$$ \mu - 9 \le X \le \mu + 9 $$

Substituting \(\sigma = 9\), this interval is:

$$ \mu - 1\sigma \le X \le \mu + 1\sigma $$

This represents the range within exactly one standard deviation of the mean.

Apply the Empirical Rule

For any normally distributed dataset, the Empirical Rule (also known as the 68-95-99.7 rule) states that:

• Approximately \(68\%\) (more precisely, \(68.27\%\)) of the data falls within one standard deviation of the mean (\(\mu \pm 1\sigma\)).
• Approximately \(95\%\) (more precisely, \(95.45\%\)) falls within two standard deviations (\(\mu \pm 2\sigma\)).
• Approximately \(99.7\%\) falls within three standard deviations (\(\mu \pm 3\sigma\)).

Since our interval is within one standard deviation, we expect approximately \(68\%\) (or \(68.27\%\) if using precise normal distribution tables) of the births to occur within this range. Standard introductory statistics curricula typically use the rounded value of \(68\%\) for the Empirical Rule. Let's state both the standard \(68\%\) and the precise \(68.27\%\) to ensure accuracy.

Answer:

The lengths of pregnancy terms for a particular species of mammal are nearly normally distributed about a mean pregnancy length with a standard deviation of 9 days. About what percentage of births would be expected to occur within 9 days of the mean pregnancy length?

About <blank>68</blank>% of births would be expected to occur within 9 days of the mean pregnancy length.
(Type an integer or a decimal. Round to two decimal places as needed.)