Question

Listed below are speeds (mi/h) measured from southbound traffic on I-280 near Cupertino, California (based on data from SigAlert). This simple random sample was obtained at 3:30pm on a weekday. The speed limit for this road is 65 mi/h

62 61 61 57 61 54 59 58 59 69 60 67

Use the sample data to construct a 95% confidence interval for the mean speed.

Answer 1

The 95% confidence interval for the mean speed is approximately
`\( (53.83, 67.49) \)` mi/h.

How did we get the value?

Given data:

`\[ n = 12 \]`

`\[ x_i \text{'s} = 62, 61, 61, 57, 61, 54, 59, 58, 59, 69, 60, 67 \]`

1. Calculate x-bar (sample mean):

`\[ \bar{x} = (62 + 61 + 61 + 57 + 61 + 54 + 59 + 58 + 59 + 69 + 60 + 67)/(12) \]`

`\[ \bar{x} = (728)/(12) \]`

`\[ \bar{x} \approx 60.66 \]`

2. Calculate
`\(s\)` (sample standard deviation):

`\[ s = \sqrt{\frac{(62-\bar{x})^2 + (61-\bar{x})^2 + \ldots + (67-\bar{x})^2}{12}} \]`

`\[ s = \sqrt{((62-60.66)^2 + (61-60.66)^2 + \ldots + (67-60.66)^2)/(12)} \]`

s = 12.08

3. Determine the Z-score for a 95% confidence interval (Z ≈ 1.96).

4. Calculate the margin of error:

`\[ \text{Margin of Error} = 1.96 \left( (s)/(√(n)) \right) \]`

`\[ \text{Margin of Error} = 1.96 \left( (12.08)/(√(12)) \right) \]`

The margin of error is approximately 6.83.

5. Construct the confidence interval:

`\[ \text{Confidence Interval} = \bar{x} \pm \text{Margin of Error} \]`

`\[ \text{Confidence Interval} = 60.66 \pm 6.83 \]`

Therefore, the 95% confidence interval for the mean speed is approximately
`\( (53.83, 67.49) \)` mi/h.