Answer :
Final answer:
The owner's level of confidence that the bull is not a carrier of the lethal gene is 97.38%. The owner needs to inseminate approximately 6 cows to be 90% confident that the bull is not a carrier.
Explanation:
In order to determine the owner's level of confidence that the bull is not a carrier of the lethal gene, we need to look at the observed results and compare them to the expected results. In this case, all 20 cows produced healthy normal calves, which is consistent with the 85% homozygous dominant and 15% heterozygous distribution in the herd. This indicates that the bull is very unlikely to be a carrier of the lethal gene. To calculate the owner's level of confidence, we can use the binomial probability formula. Given that all calves were normal, the probability of this happening by chance alone (assuming the bull is a carrier) is 0.85^20 = 0.0262. Therefore, the owner's level of confidence that the bull is not a carrier is 1 - 0.0262 = 0.9738, or 97.38%.
In question 3, to calculate the number of cows the owner needs to inseminate in order to be 90% confident that the bull is not a carrier, we can again use the binomial probability formula. We want to find the number of cows (n) that satisfies the equation 1 - (0.85^n + 0.45^n) = 0.9. Using trial and error or an iterative approach, we can find that n ≈ 6. Therefore, the owner needs to inseminate approximately 6 cows to be 90% confident that the bull is not a carrier.
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