Question

Astronauts often report episodes of disorientation as they move around the zero-gravity spacecraft. to compensate, crew members rely heavily on visual information to establish top-down orientation. An empirical study was conducted to asses the potential of using color brightness as a body orientation cue. 90 students at a specific university were randomly selected. They recline on their backs in the dark and were disorientated when postponed on a rotating platform under a slowly rotating disk that filled their entire field of vision. Half the disk was painted with a brighter level of color than the other half. The students were asked to say "stop" when they believed they were right side up. The brightness level of the disc was recorded. Of the 90 students 58 selected the brighter color level.

1) describe the parameter of interest in this study

2) calculate the parametric 90% confidence interval, checking any relevant conditions

3) interpret the interval from the previous part in the context of the problem

4) explain what is meant by 90% confidence

5) can we infer that a majority of all changes students would select brighter color levels as a body orientation cue? Explain why or why not.

Answer 1

1) describe the parameter of interest in this study

The parameter of interest in this study is the population proportion of the students who selected the brighter color level.

2) calculate the parametric 90% confidence interval, checking any relevant conditions

The confidence interval formula for the population proportion is given as below:

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Where, P is the sample proportion, Z is critical value, and n is sample size.

We are given

x = 58

n = 90

Sample size is adequate. (n>30)

P = x/n = 58/90 = 0.644444444

Confidence level = 90%

Critical Z value = 1.6449

(by using z-table)

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Confidence Interval = 0.644444444 ± 1.6449* sqrt(0.644444444*(1 – 0.644444444)/90)

Confidence Interval = 0.644444444 ± 1.6449* 0.0505

Confidence Interval = 0.644444444 ± 0.0830

Lower limit = 0.644444444 - 0.0830 = 0.5614

Upper limit = 0.644444444 + 0.0830 = 0.7274

Confidence interval = (0.5614, 0.7274)

3) interpret the interval from the previous part in the context of the problem

We are 90% confident that the population proportion of the students who selected the brighter color level will lies between 0.5614 and 0.7274.

4) explain what is meant by 90% confidence

By the 90% confidence, we are sure 90% of the time that the population proportion will lies between the given confidence interval limits.

5) can we infer that a majority of all changes students would select brighter color levels as a body orientation cue? Explain why or why not.

Yes, we can infer that a majority of all changes students would select brighter color levels as a body orientation cue, because the confidence interval have both limits greater than 0.50.