For a data set of the pulse rates for a sample of adult​ females, the lowest pulse rate is 

3535 

beats per​ minute, the mean of the listed pulse rates is 

x overbarxequals=74.074.0 

beats per​ minute, and their standard deviation is 

sequals=15.415.4 

beats per minute.

**a.** What is the difference between the pulse rate of 

3535 

beats per minute and the mean pulse rate of the​ females?

**b.** How many standard deviations is that​ [the difference found in part​ (a)]?

**c.** Convert the pulse rate of 

3535 

beats per minutes to a z score.

**d.** If we consider pulse rates that convert to z scores between 

minus−2 

and 2 to be neither significantly low nor significantly​ high, is the pulse rate of 

3535 

beats per minute​ significant?

Let's solve each part of the problem step by step.

### Given Data:
- Lowest pulse rate: \( 35 \) beats per minute
- Mean pulse rate \( \bar{x} = 74.0 \) beats per minute
- Standard deviation \( s = 15.4 \) beats per minute

### a. Difference between the pulse rate of \( 35 \) beats per minute and the mean pulse rate:

\[
\text{Difference} = 35 - 74.0 = -39.0 \text{ beats per minute}
\]

### b. Number of standard deviations that the difference represents:

\[
\text{Number of standard deviations} = \frac{\text{Difference}}{s} = \frac{-39.0}{15.4} \approx -2.53
\]

### c. Convert the pulse rate of \( 35 \) beats per minute to a z-score:

The z-score is calculated using the formula:

\[
z = \frac{X - \bar{x}}{s} = \frac{35 - 74.0}{15.4} \approx -2.53
\]

### d. Determine if the pulse rate of \( 35 \) beats per minute is significant:

If the z-score is between \( -2 \) and \( 2 \), the value is considered not significantly low or high.

\[
z = -2.53
\]

Since \( -2.53 \) is outside the range of \( -2 \) to \( 2 \), the pulse rate of \( 35 \) beats per minute **is significant**.

---

Would you like further details or have any other questions?

Here are 5 related questions to expand your understanding:
1. How do you interpret a z-score in terms of standard deviations from the mean?
2. What does it mean when a value is significant in the context of z-scores?
3. How would the z-score change if the standard deviation were smaller?
4. What is the significance of a negative z-score compared to a positive one?
5. How can z-scores be used to compare data from different distributions?

**Tip:** When interpreting z-scores, remember that a z-score tells you how many standard deviations a value is from the mean. This can be useful in determining whether a value is typical or unusual for a given data set.

Explore the analysis of pulse rates using statistics. Calculate the difference between pulse rates, determine the z-score, and assess significance based on standard deviations from the mean. Learn with examples and step-by-step solutions.

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