Question

In: Statistics and Probability

Since previous studies have reported that elite athletes are often deficient in their nutritional intake (for...

Since previous studies have reported that elite athletes are often deficient in their nutritional intake (for example, total calories, carbohydrates, protein), a group of researchers decided to evaluate Canadian high-performance athletes. A total of n = 324 athletes from eight Canadian sports centers participated in the study. One reported finding was that the average caloric intake among the n = 201 women was 2403.7 kilocalories per day (kcal/d). The recommended amount is 2811.5 kcal/d.


For one part of the study, n = 114 male athletes from eight Canadian sports centers were surveyed. Their average caloric intake was 3078.0 kilocalories per day (kcal/d) with a standard deviation of 987.0. The recommended amount is 3421.4. Is there evidence that Canadian high-performance male athletes are deficient in their caloric intake?

(a) State the appropriate H0.

i) H0: μ > 3421.4

ii) H0: μ ≠ 3421.4

iii) H0: μ = 3421.4

iv) H0: μ ≤ 3421.4

v) H0: μ < 3421.4


State the appropriate Ha.

i) Ha: μ < 3421.4

ii) Ha: μ ≠ 3421.4

iii) Ha: μ > 3421.4

iv) Ha: μ = 3421.4

v) Ha: μ ≥ 3421.4


(b) Carry out the test. (Round your answer for t to three decimal places.)

t =

Give the degrees of freedom.
df=
Give the P-value. (Round your answer to four decimal places.)
P-value=

State your conclusion.

i) We do not have sufficient evidence to conclude that Canadian high-performance male athletes are deficient in their calorie intake.

ii) We have sufficient evidence to conclude that Canadian high-performance male athletes are deficient in their calorie intake.    


(c) Construct a 95% confidence interval for the daily average deficiency in caloric intake. (Round your answers to one decimal place.)

( , ) kcal/day

Solutions

Expert Solution

Sample size n = 114

Sample mean = 3078.0 kcal/d

Sample standard deviation = s = 987.0

a ) Hypothesis :

null hypothesis

Alternative hypothesis

Left tailed test

b) test statistic :

Degrees of freedom = n - 1 = 114 - 1 = 113

P-value : P-value for this left tailed test is ,

P-value = P( t < test statistic ) = P( t < -3.715 )

Using Excel ,   =T.DIST( t , df , cumulative )

P( t < -3.715 ) = T.DIST( -3.715 , 113 , 1 ) = 0.0002

P-value = 0.0002

Decision about null hypothesis :

Rule : Reject null hypothesis if p-value less than significance level

Let ,   = 0.05

It is observed that p-value is less than   = 0.05

So reject null hypothesis

Conclusion :

We have sufficient evidence to conclude that Canadian high-performance male athletes are deficient in their calorie intake.    


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