The heights of all students at a school were measured. A mean height of 157.0 cm was calculated from
this data.
A random sample of 35 students from this school was selected. The mean height of this sample was 159.7 cm 
with a standard deviation of 8.7 cm.
The smallest confidence level that could be used to produce a confidence interval that contains μ, based on this 
sample, is

To solve this problem, we need to determine the smallest confidence level for which the calculated confidence interval still contains the population mean height (μ = 157.0 cm). We'll use the sample mean and standard deviation to calculate the confidence interval and then check when this interval just barely includes the population mean.

### Step 1: Calculate the Standard Error (SE)
The standard error of the sample mean is given by:

\[
\text{SE} = \frac{s}{\sqrt{n}}
\]

where:
- \( s = 8.7 \, \text{cm} \) (sample standard deviation),
- \( n = 35 \) (sample size).

\[
\text{SE} = \frac{8.7}{\sqrt{35}} \approx 1.47 \, \text{cm}
\]

### Step 2: Determine the Critical Value (z)
Next, we'll use the sample mean and the standard error to calculate the z-score that corresponds to the difference between the sample mean and the population mean:

\[
z = \frac{\bar{x} - \mu}{\text{SE}}
\]

where:
- \( \bar{x} = 159.7 \, \text{cm} \) (sample mean),
- \( \mu = 157.0 \, \text{cm} \) (population mean).

\[
z = \frac{159.7 - 157.0}{1.47} \approx 1.84
\]

### Step 3: Find the Corresponding Confidence Level
The z-score of 1.84 corresponds to the area under the standard normal curve. We need to find the cumulative probability for this z-score and then convert it to the confidence level.

Using the standard normal distribution table (or a calculator):

\[
P(Z \leq 1.84) \approx 0.9671
\]

This means that 96.71% of the area under the curve lies below this z-score. To find the confidence level, we calculate:

\[
\text{Confidence Level} = 2 \times 0.9671 - 1 = 0.9342
\]

Thus, the smallest confidence level that could be used to produce a confidence interval that contains the population mean \( \mu \) is approximately **93.42%**.

### Final Answer:
The smallest confidence level is **93.42%**.

Would you like more details or have any other questions?

---

Here are five related questions you might find useful:
1. How do we interpret a confidence interval in the context of a population mean?
2. What would the confidence interval be if the confidence level were set to 95%?
3. How does the sample size affect the width of a confidence interval?
4. Why is the normal distribution used in calculating confidence intervals?
5. How do we calculate the z-score for a two-tailed test?

**Tip:** Remember, a higher confidence level means a wider confidence interval, providing more certainty that the interval includes the population parameter.

The heights of all students at a school were measured. A mean height of 157.0 cm was calculated from this data. A random sample of 35 students from this school was selected. The mean height of this sample was 159.7 cm with a standard deviation of 8.7 cm. The smallest confidence level that could be used to produce a confidence interval that contains μ, based on this sample, is

Learn how to calculate the smallest confidence level that includes the population mean. Using a sample of 35 students with a mean height of 159.7 cm and standard deviation of 8.7 cm, we find the smallest confidence level for a population mean of 157.0 cm. This problem explores confidence intervals and z-scores, suitable for Grades 10-12.

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