Suppose we are interested in the mean speed of vehicles on interstate highways in the U.S. We checked the speed of 40 cars near exit 115 on the I-45 and the result is as below (in mph): 

80, 75, 65, 70, 73, 72, 79, 73, 75, 77, 80, 75, 65, 70, 73, 72, 79, 73, 75, 77, 80, 75, 65, 70, 73, 72, 79, 73, 75, 77, 80, 75, 65, 70, 73, 72, 79, 73, 75, 77. Find the lower bound of a 95% confidence interval for the proportion of cars that are over 75 mph of speed on interstate highways in the U.S. (Answer in 5 decimals.)

To find the lower bound of a 95% confidence interval for the proportion of cars that are over 75 mph, we can follow these steps:

### 1. Determine the sample proportion \( \hat{p} \):
- First, we count how many cars out of the 40 have a speed greater than 75 mph.
  
The speeds above 75 mph from the given data are:  
80, 79, 77, 80, 79, 77, 80, 79, 77, 80, 79, 77.

There are 12 cars that have a speed greater than 75 mph.

Thus, the sample proportion \( \hat{p} \) is calculated as:
\[
\hat{p} = \frac{\text{Number of cars over 75 mph}}{\text{Total number of cars}} = \frac{12}{40} = 0.30.
\]

### 2. Find the standard error (SE) of the sample proportion:
The standard error of the sample proportion is calculated using the formula:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
where \( n \) is the sample size and \( \hat{p} \) is the sample proportion.

\[
SE = \sqrt{\frac{0.30(1 - 0.30)}{40}} = \sqrt{\frac{0.30 \times 0.70}{40}} = \sqrt{\frac{0.21}{40}} = \sqrt{0.00525} \approx 0.07246.
\]

### 3. Find the z-score for a 95% confidence interval:
For a 95% confidence interval, the critical value (z-score) is 1.96.

### 4. Calculate the confidence interval:
The formula for the confidence interval is:
\[
\hat{p} \pm z \times SE
\]
where \( z \) is the z-score for the 95% confidence level.

The lower bound of the confidence interval is:
\[
\hat{p} - z \times SE = 0.30 - 1.96 \times 0.07246 \approx 0.30 - 0.14202 \approx 0.15798.
\]

Thus, the lower bound of the 95% confidence interval for the proportion of cars that are over 75 mph is approximately **0.15798**.

Let me know if you'd like more details or have any questions!

### Related Questions:
1. How would the confidence interval change with a larger sample size?
2. What if the confidence level was 90% instead of 95%? How would the interval change?
3. How do we interpret the 95% confidence interval in this context?
4. How is the standard error affected if the sample proportion is closer to 0.5?
5. What assumptions must hold for this confidence interval to be valid?

### Tip:
In confidence interval calculations, a larger sample size reduces the margin of error, providing a more precise estimate of the population parameter.

Suppose we are interested in the mean speed of vehicles on interstate highways in the U.S. We checked the speed of 40 cars near exit 115 on the I-45 and the result is as below (in mph): 80, 75, 65, 70, 73, 72, 79, 73, 75, 77, 80, 75, 65, 70, 73, 72, 79, 73, 75, 77, 80, 75, 65, 70, 73, 72, 79, 73, 75, 77, 80, 75, 65, 70, 73, 72, 79, 73, 75, 77. Find the lower bound of a 95% confidence interval for the proportion of cars that are over 75 mph of speed on interstate highways in the U.S. (Answer in 5 decimals.)

Learn how to calculate the lower bound of a 95% confidence interval for the proportion of cars traveling over 75 mph based on a sample of 40 vehicles from I-45. This problem covers key statistical concepts such as sample proportions, standard error, and confidence intervals.

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