Question

In: Statistics and Probability

Total blood volume (in ml) per body weight (in kg) is important in medical research. For...

Total blood volume (in ml) per body weight (in kg) is important in medical research. For healthy adults, the red blood cell volume mean is about μ = 28 ml/kg.† Red blood cell volume that is too low or too high can indicate a medical problem. Suppose that Roger has had seven blood tests, and the red blood cell volumes were as follows.

31 25 41 37 31 37 29

The sample mean is x ≈ 33.0 ml/kg. Let x be a random variable that represents Roger's red blood cell volume. Assume that x has a normal distribution and σ = 4.75. Do the data indicate that Roger's red blood cell volume is different (either way) from μ = 28 ml/kg? Use a 0.01 level of significance.

(a) What is the level of significance?____


State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

__H0: μ = 28 ml/kg; H1: μ < 28 ml/kg; left-tailed

__H0: μ = 28 ml/kg; H1: μ > 28 ml/kg; right-tailed    

__H0: μ = 28 ml/kg; H1: μ ≠ 28 ml/kg; two-tailed

__H0: μ ≠ 28 ml/kg; H1: μ = 28 ml/kg; two-tailed


(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

__The standard normal, since we assume that x has a normal distribution with known σ.

__The Student's t, since n is large with unknown σ.    

__The standard normal, since we assume that x has a normal distribution with unknown σ.

__The Student's t, since we assume that x has a normal distribution with known σ.


Compute the z value of the sample test statistic. (Round your answer to two decimal places.)___


(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)___

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

__At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

__At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.    

__At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

__At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.


(e) State your conclusion in the context of the application.

__There is sufficient evidence at the 0.01 level to conclude that Roger's average red cell volume differs from the average for healthy adults.

__There is insufficient evidence at the 0.01 level to conclude that Roger's average red cell volume differs from the average for healthy adults.  

Solutions

Expert Solution

Given : Sample size=n=7

Sample mean=

Population standard deviation=

Hypothesized value=

Significance level=

(a) The level of significance is

Hypothesis : ml/kg VS   ml/kg

This is two-tailed test.

(b) The standard normal , Since we assume taht X has a normal distribution with known

The Z value of the sample test statistic is ,

(c) The p-value is ,

p-value=

; From standard normal distribution table

(d) Decision : Here , p-value=0.0052 <

Therefore , reject the null hypothesis .

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

(e) Conclusion :

There is sufficient evidence at the 0.01 level to conclude that Roger's average red cell volume differs from the average for healthy adults.


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