In a survey, 25 people were asked how much they spent on their bosses last birthday gift. The results were roughly bell-shaped with a mean of $39 and standard deviation of $4. Construct and interpret a 95% confidence interval. Give your answers to one decimal place. Interpretation:
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![**Constructing a 95% Confidence Interval Based on Survey Data**
*In a survey, 25 people were asked how much they spent on their bosses' last birthday gift. The results were roughly bell-shaped with a mean of $39 and a standard deviation of $4. Construct and interpret a 95% confidence interval.*
**Give your answers to one decimal place:**
[ ] [ ]
**Interpretation:**
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### Explanation:
**Key points and formula:**
- **Sample Size (n)** = 25
- **Sample Mean (x̄)** = $39
- **Sample Standard Deviation (s)** = $4
- **Confidence Level** = 95%
Since the sample size is 25 which is less than 30, the t-distribution is preferred over the z-distribution for constructing the confidence interval.
1. **Find the t-score** associated with a 95% confidence level for 24 degrees of freedom (n-1 = 25-1 = 24).
- From the t-table, for a 95% confidence level and 24 degrees of freedom, the t-score (t*) is approximately 2.064.
2. **Calculate the Standard Error (SE)** of the mean:
\[
\text{SE} = \frac{s}{\sqrt{n}} = \frac{4}{\sqrt{25}} = \frac{4}{5} = 0.8
\]
3. **Construct the margin of error (ME)**:
\[
\text{ME} = t^* \times \text{SE} = 2.064 \times 0.8 = 1.6512
\]
4. **Determine the confidence interval**:
- **Lower limit**: \( x̄ - \text{ME} = 39 - 1.6512 = 37.3 \)
- **Upper limit**: \( x̄ + \text{ME} = 39 + 1.6512 = 40.7 \)
Thus, the 95% confidence interval is [37.3, 40.7].
**Interpretation:**
We are 95% confident that the true mean amount spent on bosses' last birthday gifts by the surveyed group lies between $37.3 and $40.7.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9364916e-ff41-4b95-a11d-57a67c815768%2Fcd9d0447-131e-44a2-8e02-9ef51b032776%2Fx0trqr6_processed.jpeg&w=3840&q=75)
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