Emilio has played ten rounds of golf this season. His mean score is 80 and standard deviation is 4. Assume Emilio's golf scores are normally distributed.
a) Find the 95% confidence interval of Emilio's mean golf score
b)Find the 95% confidence interval of Emilio's mean golf score if the standard deviation was 8 instead of 4
c) Emilio's most recent golf score is 75. He claims that his game has improved and hid latest score should determine whether he qualifies for entry in the tournament, should the tournament accept his claim?
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To find the 95% confidence interval of Emilio's mean golf score, we can use the formula:
Confidence Interval = Mean ± Critical Value × (Standard Deviation / √Sample Size)
where Mean is the sample mean, Standard Deviation is the population standard deviation, and Sample Size is the number of observations.
a) Given that Emilio has played ten rounds of golf with a mean score of 80 and a standard deviation of 4, we can calculate the confidence interval as follows:
Mean = 80
Standard Deviation = 4
Sample Size = 10
Critical Value (z-value for 95% confidence interval) ≈ 1.96 (from z-table)
Confidence Interval = 80 ± 1.96 × (4 / √10)
Now let's calculate the confidence interval:
Confidence Interval = 80 ± (1.96 × 4 / √10)
Confidence Interval ≈ 80 ± 2.46
Therefore, the 95% confidence interval of Emilio's mean golf score is approximately (77.54, 82.46).
b) If the standard deviation were 8 instead of 4, we would repeat the calculation with the new standard deviation:
Mean = 80
Standard Deviation = 8
Sample Size = 10
Critical Value (z-value for 95% confidence interval) ≈ 1.96
Confidence Interval = 80 ± 1.96 × (8 / √10)
Confidence Interval ≈ 80 ± 4.92
Therefore, the 95% confidence interval of Emilio's mean golf score with a standard deviation of 8 would be approximately (75.08, 84.92).
c) To determine if Emilio's latest score of 75 qualifies him for entry in the tournament, we need to check if it falls within the 95% confidence interval of his mean golf score. Since his most recent score is lower than the lower bound of the confidence interval (77.54), it suggests that his game has improved significantly. Therefore, based on his claim and the data provided, it is reasonable to accept his claim and allow him entry into the tournament.