Question

A research group is curious about features that can be attributed to music genres. A music streaming service provides a few different attributes for songs such as speechness, danceability, and valence. They suspect that there is a difference between the average valence (positive or negative emotion) of metal songs compared to blues songs. However, they must conduct a study to determine if that is true. From a sample of 87 metal songs, the sample mean for valence is 0.451 and the sample standard deviation is 0.139. From a sample of 94 blues songs, the sample mean for valence is 0.581 and the sample standard deviation is 0.167.

Assume that sample1 comes from the sample metal songs and that sample2 comes from the sample blues songs.

Compute the 90% confidence interval. Please round the values to the fourth decimal point and format your response as follows: (lower_value, upper_value),

Answer 1

Explanation:

Hello!

The researchers are interested in estimating the difference between the average valence of metal songs and the average valence of blues songs.

To do so, two independent samples were taken:

Sample 1

X₁: Valence of a heavy metal song.

n₁= 87 metal songs

X[bar]₁= 0.451

S₁= 0.139

Sample 2

X₂: Valence of a blues song.

n₂= 94 blues songs

X[bar]₂= 0.581

S₂= 0.167

There is no information about the distribution of the variables but since both samples sizes are big enough (both are higher than 30) I'll apply the Central Limit Theorem and approximate the distribution of both samples means to normal:

X[bar]₁ ≈N(μ₁;δ₁²/n₁)

X[bar]₂ ≈N(μ₂;δ₂²/n₂)

Now it is valid to use an approximation to the standard normal to calculate the 90% confidence interval:

[(X[bar]₁ - X[bar]₂) ±
`Z_(1-\alpha /2)` *
`\sqrt{(S^2_1)/(n_1)+(S^2_2)/(n_2) }`]

[(0.451-0.581)±1.648*
`\sqrt{(0.139^2)/(87)+(0.167^2)/(94) }`]

[-0.168;-0.093]

With a confidence level of 90%, you'd expect that the interval [-0.168;-0.093] will contain the difference between the population mean of the valence of heavy metal songs and the population mean of the valence of blues songs.

I hope it helps!