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Biologists researching the effects of adding limestone sand as buffer for acid rain effects in streams monitored the pH levels of two streams each month for 36 months. The first stream had a mean pH level of 6.8 with a standard deviation of 2.3. The control stream had a mean pH level of 9.2 with a standard deviation of 1.5. Assume a 0.05 significance level for testing the claim that the mean pH of the first stream was less (more acidic) than the mean pH of the control stream. Also, assume the two samples are independent simple random samples selected from normally distributed populations. Construct a 90% confidence interval for the difference between the two means. i. Calculate the sample mean. ii. Calculate the standard error. iii. Find the critical value. iv. Construct the confidence interval. v. Interpret the confidence interval.

User Maweeras
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Answer:

Step-by-step explanation:

To solve this problem, we'll follow the steps you've outlined:

i. Calculate the sample mean:

The sample mean of the first stream is given as 6.8, and the sample mean of the control stream is 9.2.

ii. Calculate the standard error:

The standard error can be calculated using the formula:

SE = sqrt((s1^2 / n1) + (s2^2 / n2))

where s1 and s2 are the standard deviations of the first and control streams, respectively, and n1 and n2 are the sample sizes.

For the first stream:

s1 = 2.3 (standard deviation)

n1 = 36 (sample size)

For the control stream:

s2 = 1.5 (standard deviation)

n2 = 36 (sample size)

SE = sqrt((2.3^2 / 36) + (1.5^2 / 36))

iii. Find the critical value:

Since the significance level is 0.05 and we're conducting a one-tailed test (claiming that the mean pH of the first stream is less), we need to find the critical value from the t-distribution. With a sample size of 36 - 1 = 35, the degrees of freedom (df) are 35. Using a t-table or calculator, we find the critical value for a 0.05 significance level and 35 degrees of freedom is approximately -1.692.

iv. Construct the confidence interval:

The formula for a confidence interval for the difference between two means is:

CI = (x1 - x2) ± (critical value * SE)

where x1 and x2 are the sample means.

CI = (6.8 - 9.2) ± (-1.692 * SE)

v. Interpret the confidence interval:

The confidence interval will provide a range of values within which we can be 90% confident that the true difference between the means of the first and control streams lies. Specifically, we are constructing a 90% confidence interval for the difference between the mean pH levels of the two streams.

User Kitchenprinzessin
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