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Let x be a random variable that represents the averagedaily temperature (in degrees Fahrenheit) in July in a town inColorado. The x distribution has a mean μ ofapproximately 75°F and standard deviation σ ofapproximately 8°F. A 20-year study (620 July days) gave the entriesin the rightmost column of the following table.

I       Â	II	III	IV
Region under Normal Curve	x°F	Expected % from Normal Curve	Observed Number of Days in 20 Years
μ – 3σ ≤ x < μ –2σ	  51 ≤ x < 59	2.35%         Â	12           Â
μ – 2σ ≤ x < μ–  σ	  59 ≤ x < 67	13.5%         Â	90           Â
μ –  σ ≤ x <μ	  67 ≤ x < 75	34%         Â	206           Â
μ ≤ x < μ + σ	  75 ≤ x < 83	34%         Â	215           Â
μ + σ ≤ x < μ +2σ	  83 ≤ x < 91	13.5%         Â	82           Â
μ + 2σ ≤ x < μ +3σ	  91 ≤ x < 99	2.35%         Â	15           Â

(i) Remember that μ = 75 and σ = 8. Examinethe figure above. Write a brief explanation for columns I, II, andIII in the context of this problem.

This answer has not been graded yet.

(ii) Use a 1% level of significance to test the claim that theaverage daily July temperature follows a normal distribution withμ = 75 and σ = 8.(a) What is the level ofsignificance?


State the null and alternate hypotheses.

H₀: The distributions are different.
H₁: The distributions aredifferent.H₀: The distributions aredifferent.
H₁: The distributions are thesame.     H₀: Thedistributions are the same.
H₁: The distributions are thesame.H₀: The distributions are the same.
H₁: The distributions are different.

(b) Find the value of the chi-square statistic for the sample.(Round the expected frequencies to at least three decimal places.Round the test statistic to three decimal places.)


Are all the expected frequencies greater than 5?

YesNo     

What sampling distribution will you use?

normalbinomial     Student'stchi-squareuniform

What are the degrees of freedom?


(c) Estimate the P-value of the sample test statistic.

P-value > 0.1000.050 < P-value <0.100     0.025 < P-value <0.0500.010 < P-value < 0.0250.005 <P-value < 0.010P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject orfail to reject the null hypothesis that the population fits thespecified distribution of categories?

Since the P-value > α, we fail to rejectthe null hypothesis.Since the P-value > α, wereject the null hypothesis.     Since theP-value ≤ α, we reject the null hypothesis.Sincethe P-value ≤ α, we fail to reject the nullhypothesis.

(e) Interpret your conclusion in the context of theapplication.

At the 1% level of significance, the evidence is sufficient toconclude that the average daily July temperature does not follow anormal distribution.At the 1% level of significance, the evidenceis insufficient to conclude that the average daily July temperaturedoes not follow a normal distribution.