(a) To calculate the 95% confidence interval for the population correlation, we can use Fisher's Z transformation.

First, we need to calculate the Z value for the given correlation coefficient using the following formula:

Z = (1/2) * ln((1+r)/(1-r))
where r is the correlation coefficient.

Z = (1/2) * ln((1+0.27)/(1-0.27))
Z = (1/2) * ln(1.27/0.73)
Z = (1/2) * ln(1.7397)
Z ≈ 0.6531

Next, we need to find the critical Z value for a 95% confidence interval, which corresponds to a two-tailed test. Since the sample size is large (n > 30), we can use the standard normal distribution.

The critical Z value for a 95% confidence interval is approximately 1.96.

Now, we can calculate the confidence interval:

Z ± (1.96 * standard error of Z)
= 0.6531 ± (1.96 * (1/sqrt(n-3)))
= 0.6531 ± (1.96 * (1/sqrt(350-3)))
= 0.6531 ± (1.96 * (1/sqrt(347)))
= 0.6531 ± (1.96 * 0.05368)
≈ 0.6531 ± 0.10513

Therefore, the 95% confidence interval for the population correlation is approximately 0.548 to 0.758.

(b) To calculate the p-value for testing if the population correlation is zero, we can use the following formula:

p-value = 2 * (1 - Φ(|Z|))
where Φ(|Z|) is the cumulative distribution function of the standard normal distribution at |Z|.

In this case, |Z| = |0.6531| = 0.6531.

p-value = 2 * (1 - Φ(0.6531))

Looking up the value of Φ(0.6531) in a standard normal distribution table or using a calculator, we find the value to be approximately 0.7425.

p-value = 2 * (1 - 0.7425)
≈ 2 * 0.2575
= 0.515

Therefore, the p-value for testing if the population correlation is zero is approximately 0.515.

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