Question

Overweight participants who lose money when they don’t meet a specific exercise goal meet the goal more often, on average, than those who win money when they meet the goal, even if the final result is the same financially. In particular, participants who lost money met the goal for an average of 45.0 days (out of 100) while those winning money or receiving other incentives met the goal for an average of 33.7 days. The incentive does make a difference. In this exercise, we ask how big the effect is between the two types of incentives. Find a 90% confidence interval for the difference in mean number of days meeting the goal, between people who lose money when they don't meet the goal and those who win money or receive other similar incentives when they do meet the goal. The standard error for the difference in means from a bootstrap distribution is 4.14.

Answer 1

The 90% confidence interval for the difference in mean number of days meeting the goal is (4.49, 18.11).

Explanation:

The (1 - α)% confidence interval for the difference between two means is:

`CI=\bar x_(1)-\bar x_(2)\pm z_(\alpha/2)* SE_{\text{diff}}`

It is provided that:

`\bar x_(1)=45\\\bar x_(2)=33.7\\SE_{\text{diff}} =4.14\\\text{Confidence Level}=90\%`

The critical value of z for 90% confidence level is,

z = 1.645

*Use a z-table.

Compute the 90% confidence interval for the difference in mean number of days meeting the goal as follows:

`CI=\bar x_(1)-\bar x_(2)\pm z_(\alpha/2)* SE_{\text{diff}}`

`=45-33.7\pm 1.645* 4.14\\\\=11.3\pm 6.8103\\\\=(4.4897, 18.1103)\\\\\approx (4.49, 18.11)`

Thus, the 90% confidence interval for the difference in mean number of days meeting the goal is (4.49, 18.11).