To calculate the percentage of people with readings between 156 and 174, we will:

1. Convert the given readings (156 and 174) to z-scores:

   • Use the formula: $z = \frac{x - \mu}{\sigma}$ where:
     • $x$ = raw score,
     • $\mu = 124$ (mean),
     • $\sigma = 20$ (standard deviation).

2. Use the table to find the cumulative percentages for the corresponding z-scores.

3. Subtract the smaller cumulative percentage from the larger cumulative percentage to find the percentage of people within the range.

Step 1: Calculate the z-scores

• For $x = 156$: $z = \frac{156 - 124}{20} = \frac{32}{20} = 1.6$
• For $x = 174$: $z = \frac{174 - 124}{20} = \frac{50}{20} = 2.5$

Step 2: Lookup Percentiles in the Table

• From the table:
  • At $z = 1.6$, the percentile is $94.52\%$,
  • At $z = 2.5$, the percentile is $99.38\%$.

Step 3: Calculate the Percentage

The percentage of people with readings between 156 and 174 is: $99.38\% - 94.52\% = 4.86\%$

Final Answer:

The percentage of people with readings between 156 and 174 is 4.9% (rounded to the nearest tenth).

Let me know if you need further clarifications or additional details! Here are some follow-up questions:

1. How are z-scores used in other statistical contexts?
2. What is the significance of cumulative percentages in probability?
3. How does changing the standard deviation affect the z-scores and percentages?
4. What does a z-score of 0 represent in a normal distribution?
5. Can this method be applied to non-normal distributions?

Tip: Always double-check the z-score table for cumulative percentages to ensure accurate calculations!