MS 150 Statistics quiz 10 hyp test • Name:

The hue angle color system assigns an angle for each color from red to violet in the spectrum. Red is 0° green is 120° and blue is 240°. A photograph of a rainbow taken on Pohnpei was analyzed and the color angles for the rainbow were measured and recorded as seen in the image.

9 30 64 111 137 165 184 239 270 303

Data: 9;17;24;26;30;36;43;48;52;064;077;088;111;137;165;184;200;217;230;236;239;250;270;303
The data above can be copied and pasted into OpenOffice.org functions to verify the mean and standard deviation.

Hypothesis test one

For the rainbow hue angle data, the sample size n is 24, the sample mean x is 127.33, and the sample standard deviation is 95.89.

Newton put green at the center of his red-orange-yellow-green-blue-indigo-violet rainbow. On a color wheel green is at 120° . Test the hypothesis that the population mean μ hue angle for a rainbow is 120°, in other words, that the middle of the rainbow is actually green.

Null hypothesis: H0: μ = 120°
Alternate hypothesis: H1: μ ≠ 120°

  1. Calculate the t-statistic. t = __________
  2. Determine the p-value using the t-distribution. p-value = __________
  3. ________________________________________ Would we reject the null hypothesis or fail to reject the null hypothesis that the mean hue angle of 120° is statistically significantly different from the mean in question three at a 5% level of significance?
  4. __________ If we reject the null hypothesis, what is the risk of a type I error based on the p-value?
  5. __________ What is the maximum confidence level c for which this green is not the middle of the rainbow?
  6. __________ Could green be the middle of the rainbow at an alpha α = 0.05?

Hypothesis test two

On a 360° color wheel the mean hue angle is cyan at 180°. Test the hypothesis that the population mean μ hue angle for a rainbow is 180°, in other words, that the middle of the rainbow is actually cyan.

Null hypothesis: H0: μ = 180°
Alternate hypothesis: H1: μ ≠ 180°

  1. Calculate the t-statistic. t = __________
  2. Determine the p-value using the t-distribution. p = __________
  3. ________________________________________ Would we reject the null hypothesis or fail to reject the null hypothesis that the mean hue angle of 180° degrees is statistically significantly different from the mean in question three at a 5% level of significance?
  4. __________ If we reject the null hypothesis, what is the risk of a type I error based on the p-value?
  5. __________ What is the largest confidence interval c for which this difference is statistically significant?
  6. __________ Could cyan be the middle of the rainbow at an alpha α = 0.05?

Formulas are written for OpenOffice.org Calc. Replace semi-colons with commas for Excel.

Hypothesis testing for a sample mean versus a known population mean
Statistic or ParameterSymbolEquationsOpenOffice
Relationship between confidence level c and alpha α for two-tailed tests 1 − c = α
Calculate t-critical for a two-tailed test tc=TINV(α;df)
Calculate a t-statistic t t-statistic =(x - μ)/(sx/SQRT(n))
Calculate a two-tailed p-value from a t-statisticp-value = TDIST(ABS(t);df;2)

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