**Example 4.3: Changing the Confidence Level**

Suppose we change the original problem by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.

**Solution**

To find the confidence interval, you need the sample mean, , and
the EBM.

σ =3 ; n = 36 ; The confidence level is 95% (CL = 0.95)

CL = 0.95 so α = 1 − CL = 1 − 0.95 = 0.05

The area to the right of z_{.025} is 0.025 and the area to the left of z_{.025} is 1−0.025 = 0.975

using invnorm(.975,0,1) on the TI-83,83+,84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the
Standard Normal distribution.)

**Interpretation**

We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.

**Explanation of 95% Confidence Level**

95% of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score.

**Comparing the results**

The 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider.

**Summary: Effect of Changing the Confidence Level**

- Increasing the confidence level increases the error bound, making the confidence interval wider.
- Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

**Example 4.4: Changing the Sample Size:**

Suppose we change the original problem to see what happens to the error bound if the sample size is changed.

See the following Problem.

**Problem**

Leave everything the same except the sample size. Use the original 90% confidence level. What happens to the error bound and the confidence interval if we increase the sample size and use n=100 instead of n=36? What happens if we decrease the sample size to n=25 instead of n=36?

- σ = 3 ; The confidence level is 90% (CL = 0.90) ;

**Solution A**

If we **increase** the sample size n to 100, we **decrease** the error bound.

When n = 100 :

**Solution B**

If we **decrease** the sample size n to 25, we **increase** the error bound.

When n = 25 :

**Summary: Effect of Changing the Sample Size**

- Increasing the sample size causes the error bound to decrease, making the confidence interval narrower.
- Decreasing the sample size causes the error bound to increase, making the confidence interval wider.

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