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Since previous studies have reported that elite athletes are often deficient in their nutritional intake, 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 kcal/day. The recommended amount is 2811.5 kcal/day. Is there evidence that female Canadian high performance athletes are deficient in caloric intake?

(a) State the appropriate H0 and HA to test this.

(b) Assuming a standard deviation of 880 kcal/day, carry out the test at the α = 0.05 level. Make sure to interpret the results in plain language.

Now suppose a nutritionist is brought in to implement a new health program for the female athletes. This program should increase mean caloric intake but not change the standard deviation. Given the standard deviation, and how calorically deficient these athletes are, a change in the mean of 50 kcal/day to 2453.7 is of little practical importance. However, with a large enough sample, this change can be statistically significant. To see this, calculate the P-value for the test of H0 : µ = 2403.7 versus HA : µ > 2403.7. in each of the following situations:

c) In a sample of 500 athletes, the average caloric intake is x¯ = 2453.7.

(d) In a sample of 2500 athletes, the average caloric intake is x¯ = 2453.7.