Question

Listening to music has long been thought to enhance intelligence, especially during infancy and childhood. To test whether this is true, a researcher records the number of hours that eight high-performing students listened to music per day for 1 week. The data are listed in the table.

Music Listening Per Day (in hours)
4.2
4.7
4.9
3.7
4.2
5.6
4.0
4.3

(a) Find the confidence limits at a 95% CI for this one-independent sample. (Round your answers to two decimal places.)
to hours per day

Answer 1

Given Data (n = 8):

$$4.2,4.7,4.9,3.7,4.2,5.6,4.0,4.3$$

Step 1: Calculate the Sample Mean ($\bar{X}$)

$$\bar{X}=\frac{4.2+4.7+4.9+3.7+4.2+5.6+4.0+4.3}{8}=\frac{35.6}{8}=4.45\text{hours}$$

Step 2: Calculate the Sample Standard Deviation ($s$)

$$s=\sqrt{\frac{\sum (X_i-\bar{X}{)}^2}{n-1}}$$

Deviations from mean:

$$(-0.25,0.25,0.45,-0.75,-0.25,1.15,-0.45,-0.15)$$

Squared deviations:

$$0.0625,0.0625,0.2025,0.5625,0.0625,1.3225,0.2025,0.0225$$

Sum of squared deviations:

$$2.5$$$$s=\sqrt{\frac{2.5}{7}}\approx 0.60\text{hours}$$

Step 3: Determine the t-Score (95% CI, df = 7)

From t-table:

$$t_{0.025,7}\approx 2.365$$

Step 4: Compute the Margin of Error (ME)

$$ME=t\times \left(\frac{s}{\sqrt{n}}\right)=2.365\times \left(\frac{0.60}{\sqrt{8}}\right)\approx 0.50\text{hours}$$

Step 5: Calculate the 95% Confidence Interval

$$CI=\bar{X}\pm ME=4.45\pm 0.50=(3.95,4.95)\text{hours}$$

Answer:

The 95% confidence interval for music listening hours per day is 3.95 to 4.95 hours.

Answer 2

The statistic software output for this problem is :

A 95% CI is :

3.95 to 4.95 hours per day