Question

Let x be a random variable that represents red blood cell count (RBC) in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal. For the population of healthy female adults, suppose the mean of the x distribution is about 4.66. Suppose that a female patient has taken six laboratory blood tests over the past several months and that the RBC count data sent to the patient's doctor are as follows.

4.9

4.2

4.5

4.1

4.4

4.3

(i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.)

x	=
s	=

(ii) Do the given data indicate that the population mean RBC count for this patient is lower than 4.66? Use α = 0.05.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ > 4.66; H₁: μ = 4.66H₀: μ = 4.66; H₁: μ < 4.66    H₀: μ < 4.66; H₁: μ = 4.66H₀: μ = 4.66; H₁: μ ≠ 4.66H₀: μ = 4.66; H₁: μ > 4.66

(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 and σ is unknown.The Student's t, since we assume that x has a normal distribution and σ is unknown.    The Student's t, since we assume that x has a normal distribution and σ is known.The standard normal, since we assume that x has a normal distribution and σ is known.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(c) Estimate the P-value.

P-value > 0.2500.100 < P-value < 0.250    0.050 < P-value < 0.1000.010 < P-value < 0.050P-value < 0.010

Sketch the sampling distribution and show the area corresponding to the P-value.

(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 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

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

There is sufficient evidence at the 0.05 level to conclude that the population mean RBC count for the patient is lower than 4.66.There is insufficient evidence at the 0.05 level to conclude that the population mean RBC count for the patient is lower than 4.66.    

Answer 1

Given that,
i.
population mean(u)=4.66
sample mean, x =4.4
standard deviation, s =0.2828
number (n)=6
null, Ho: μ=4.66
alternate, H1: μ<4.66
level of significance, α = 0.05
from standard normal table,left tailed t α/2 =2.015
since our test is left-tailed
reject Ho, if to < -2.015
we use test statistic (t) = x-u/(s.d/sqrt(n))
to =4.4-4.66/(0.2828/sqrt(6))
to =-2.252
| to | =2.252
critical value
the value of |t α| with n-1 = 5 d.f is 2.015
we got |to| =2.252 & | t α | =2.015
make decision
hence value of | to | > | t α| and here we reject Ho
p-value :left tail - Ha : ( p < -2.252 ) = 0.03705
hence value of p0.05 > 0.03705,here we reject Ho
ANSWERS
---------------
a.
level of significance =0.05
b.
The Student's t, since we assume that x has a normal distribution and σ is unknown.
null, Ho: μ=4.66
alternate, H1: μ<4.66
test statistic: -2.252
critical value: -2.015
decision: reject Ho
c.
p-value: 0.03705
d.
At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
we have enough evidence to support the claim that the population mean RBC count for this patient is lower than 4.66
e.
There is sufficient evidence at the 0.05 level to conclude that the population mean RBC count for the patient is lower than 4.66.