Question

Music has long been known to influence consumer behavior. Many retail establishments, from Starbucks to Safeway, choose the music they play for shoppers strategically as part of their larger marketing and branding strategies as well as a tool to encourage patrons to spend money. Factors such as the tempo (or speed), pitch, and key of a song have differing effects on behavior. You can read more about this issue here if you're curious (https://www.psychologytoday.com/blog/your-musical-self/201412/music-and-the-shopper) if you are interested. You won't need any outside information to work this problem, however.

The Muzak company provides "background music solutions" to businesses in a variety of industries, including auto dealerships, salons and spas, and restaurants. In a recent study, researchers wanted to see what effect a song's tempo might have on alcohol sales (in dollars) at restaurants.

One restaurant was observed on both Fridays and Saturdays for six weeks. For three weeks, fast music (Fast) was played on Fridays and slow music (Slow) was played on Saturdays, and for another three weeks the pattern was reversed. At the end of each night, researchers examined the bar charges on each check. Because a check could contain charges for multiple people, each observation in the study was called a "party." At the end of the study, bar charges for 1,200 parties had been collected. The means and standard deviations of alcohol sales (both in dollars) broken down by party are shown below.

Music	Mean	Standard Deviation	n
Slow (Group 1)	67.51	8.50	590
Fast (Group 2)	47.90	7.87	610

Calculate and interpret a 95% confidence interval for the difference in mean alcohol sales between parties exposed to slow music and those exposed to fast music. Assume the population standard deviations are not equal (Case 2). Read carefully, and take all calculations toward the answer to three (3) decimal places.

		The researchers can be 95% confident that the true difference in mean alcohol sales between when the music is slow and when the music is fast is between $-41.47 and $20.54. They cannot conclude that the tempo of the music has an effect on average alcohol sales because zero lies within the interval.
		The researchers can be 95% confident, that the true difference in mean alcohol sales between when the music is slow and when the music is fast is between $-41.47 and $20.83. They cannot conclude that the tempo of the music has an effect on average alcohol sales because zero lies within the interval.
		The researchers can be 95% confident that the true difference in mean alcohol sales between when the music is slow and when the music is fast is between $-1.96 and $1.96. They cannot conclude that the tempo of the music has an effect on average alcohol sales because zero lies within the interval.
		The researchers can be 95% confident that the true difference in mean alcohol sales between when the music is slow and when the music is fast is between $-18.83 and $20.39. They cannot conclude that the tempo of the music has an effect on average alcohol sales because zero lies within the interval.
		The researchers can be 95% confident that the true difference in mean alcohol sales between when the music is slow and when the music is fast is between $18.68 and $20.54. They can conclude that alcohol sales are higher on average when slow music is played than when fast music is played.
		The researchers can be 95% confident that the true difference in mean alcohol sales between when the music is slow and when the music is fast is between $18.40 and $20.83. They can conclude that alcohol sales are higher on average when fast music is played than when slow music is played.
		The researchers can be 95% confident that the true difference in mean alcohol sales when the music is slow and when the music is fast is between $-20.54 and $-18.68. They can conclude that alcohol sales are lower on average when slow music is played than when fast music is played.

Answer 1

Solution

95% confidence interval = (sample mean slow - sample mean fast +-1.96(²_slow/n_slow + ²_fast/n_fast)^0.5

= ((67.51-47.90)+-1.96(8.5²/590 + 7.87²/610)^0.5 )

95% CI = (18.68, 20.54)

The researchers can be 95% confident that the true difference in mean alcohol sales between when the music is slow and when the music is fast is between $18.68 and $20.54. They can conclude that alcohol sales are higher on average when slow music is played than when fast music is played.