The manufacturer of an MP3 player wanted to know whether a...

The manufacturer of an MP3 player wanted to know whether a 10 percent reduction in price is enough to increase the sales of their product. To investigate, the owner randomly selected eight outlets and sold the MP3 player at the reduced price. At seven randomly selected outlets, the MP3 player was sold at the regular price. Reported below is the number of units sold last month at the sampled outlets. At the .01 significance level, can the manufacturer conclude that the price reduction resulted in an increase in sales?

Key Concepts

One-Tailed Test

A one-tailed test is used when the research question predicts the direction of an effect, for example, whether a reduction in price leads to higher sales. This type of test focuses on the possibility of the effect in one specific direction, making it appropriate when the hypothesis specifically predicts an increase in sales.

Two-Sample t-Test

A two-sample t-test is a statistical test used to determine whether there is a significant difference between the means of two independent groups. It is particularly useful when comparing the effects of different treatments or conditions, such as a price reduction vs. a regular price, and accounts for variability in the sample data.

Null and Alternative Hypotheses

The null hypothesis (H0) typically represents a statement of no effect or no difference, while the alternative hypothesis (Ha) represents the claim to be tested, such as an increase in sales due to a price reduction. Formulating these hypotheses clearly is critical as they guide the analysis and interpretation of results.

Hypothesis Testing

Hypothesis testing is a statistical method that involves making an assumption (the null hypothesis) about a population parameter and then using sample data to assess whether there is enough evidence to reject that assumption in favor of an alternative hypothesis. It provides a structured process to decide whether observed effects are statistically significant or could have occurred by chance.

Significance Level

The significance level, often denoted by alpha (?), is the probability threshold for rejecting the null hypothesis. It defines the risk level of making a Type I error, which is rejecting the null hypothesis when it is actually true. In this context, the significance level is set at 0.01, meaning that there is a 1% risk of concluding that the price reduction increased sales when it did not.

Transcript

00:01 So we're going to have a 1 % significance test to determine if reducing the price seems to increase sales.

00:08 So we'll have group one will be the regular price and group two will be the reduced sale 10 % off.

00:16 And they want to see does that reduction cause an increase.

00:19 So we have more sales with that second group that is reduced.

00:23 And we have some sales figures and we have from our first group had a sample size of, of seven.

00:32 And our second group had a sample size for the reduced value of eight.

00:35 And we had a mean for the first group.

00:38 The mean sales for the first group was 117 .714 with a sample standard deviation of 19 .914.

00:48 And for our second group, they had a mean sales of 125 .125.

00:55 And the sample standard deviation was 15 .094.

01:03 Now, we will be doing a two sample t test due to the fact that we have small sample sizes that are less than 30, and we don't know that these populations are normal.

01:16 So we'll have to use the t test.

01:18 We're also going to be assuming that these two standard deviations are approximately equal, and that causes us to have to find the pooled variance.

01:27 And our pooled variance we'll determine is to take one less than the first sample size times the variance of the first group, and one less than the sample size times the variance of the second group, and then divided by actually our degrees of freedom, which is 7 plus 8 minus 2.

01:46 So that's 15, that's 13.

01:47 So our degrees of freedom will be 13.

01:50 So let's get this calculation and go from there.

01:55 So we have left parenthesis that six times that 19 .914 squared.

02:01 And let me do a little correction there.

02:03 Just hit something wrong.

02:05 Okay, plus the seven times the 15 .094 squared.

02:12 Close off that parentheses and then divided by my 13...