The Standard Deviation Of Sample Means Is The Same As The Po

The Standard Deviation Of Sample Means Is The Same As The Population S

The standard deviation of sample means is the same as the population standard deviation. The distribution of sample means (x-bar values) for large random samples follows a bell-shaped curve only if the individual population values follow a normal distribution. (Hint: see the “Conditions for the Rule for Sample Means”)

“A 95% confidence interval for the mean weight loss for men is 6.4 to 11.2 pounds.” This means that 95% of all men will lose between 6.4 and 11.2 pounds. If a 95% confidence interval calculated for the difference between two population means is –4.31 to 0.76, then we may conclude with high confidence that the two population means have different values. The weights for a population of North American raccoons has a bell-shaped frequency curve with a mean of about 12 pounds and a standard deviation of about 2.5 pounds.

About 95% of the weights for individual raccoons in this population fall between what two values? The weights for a population of North American raccoons has a bell-shaped frequency curve with a mean of about 12 pounds and a standard deviation of about 2.5 pounds. About 95% of the weights for individual raccoons in this population fall between what two values? The weights for a population of North American raccoons has a bell-shaped frequency curve with a mean of about 12 pounds and a standard deviation of about 2.5 pounds. About 95% of the weights for individual raccoons in this population fall between what two values?

The Baltimore Sun (Haney, 21 February 1995) reported on a study by Dr. Sara Harkness, in which she observed the sleep patterns of 6-month-old infants. She found that a sample of n = 49 infants in the U.S. slept an average of 13 hours per day, and that the standard deviation of this sample was 0.5 hours. Compute a 90% confidence interval for the mean sleep time per day for 6-month-old infants in the U.S. Show your work and express your answer using exactly two (2) decimal places.

The Baltimore Sun (Haney, 21 February 1995) reported on a study by Dr. Sara Harkness, in which she observed the sleep patterns of 6-month-old infants. She found that a sample of n = 49 infants in the U.S. slept an average of 13 hours per day, and that the standard deviation of this sample was 0.5 hours. Interpret the confidence interval you computed above using words that someone with no training in statistics might understand.

Paper For Above instruction

The concept of the standard deviation of sample means, also known as the standard error, is pivotal in understanding the relationship between sample statistics and population parameters in inferential statistics. The statement that "The standard deviation of sample means is the same as the population standard deviation" underscores that, under certain conditions, the variability of the sample means (derived from different samples) is directly related to the population variability divided by the square root of the sample size (n). Specifically, for large random samples drawn from a normally distributed population, the distribution of the sample means will approximate a normal distribution, with its standard deviation (the standard error) calculated as the population standard deviation divided by √n.

This principle is integral to constructing confidence intervals—ranges within which we expect the true population parameter to lie with a certain level of confidence. For instance, a 95% confidence interval for the mean weight loss in men, spanning from 6.4 to 11.2 pounds, suggests that if we were to take many samples and compute such intervals, approximately 95% of these intervals would contain the true average weight loss for all men. It's important to note that this interpretation does not mean that 95% of individual men will lose weights within this range, but rather that the method used to create the interval has a 95% success rate in capturing the true mean across numerous samples.

Similarly, when analyzing differences between two population means, a confidence interval that spans from –4.31 to 0.76 suggests high confidence that the actual difference in population means is within this range. Because this interval includes zero, it indicates that there might be no difference at all, but given the context, it still reflects significant uncertainty.

Turning to the example of North American raccoons, with a population mean weight of about 12 pounds and a standard deviation of 2.5 pounds, we can determine the range covering approximately 95% of individual raccoon weights by applying the empirical rule. For a bell-shaped distribution, roughly 95% of individual data points fall within two standard deviations of the mean. Thus, 12 ± 2*2.5 = (12 – 5, 12 + 5) = (7, 17). Therefore, approximately 95% of individual raccoons weigh between 7 and 17 pounds.

When considering the sample means from groups of 25 raccoons, the variability of these means—the standard error—is smaller because it accounts for the distribution across many samples. For such samples, the standard error is found by dividing the population standard deviation by √n, which is √25 = 5. Therefore, the standard error = 2.5 / 5 = 0.5 pounds. Applying the empirical rule again, 95% of the sample means from these groups will fall within 12 ± 2*0.5 = (11, 13). This indicates that if many samples of 25 raccoons are taken, the average weight of each sample will likely fall between 11 and 13 pounds 95% of the time.

The study conducted by Dr. Sara Harkness on 6-month-old infants provides a practical application of confidence intervals in real-world research. With a sample size of 49 infants, an average sleep time of 13 hours, and a standard deviation of 0.5 hours, a 90% confidence interval can be constructed to estimate the mean sleep duration for all such infants.

To compute this, the standard error is calculated as the sample standard deviation divided by the square root of sample size: 0.5 / √49 = 0.5 / 7 ≈ 0.0714 hours. The critical z-value for a 90% confidence interval (α=0.10) is approximately 1.645. The margin of error (ME) is then 1.645 * 0.0714 ≈ 0.1176 hours.

Therefore, the 90% confidence interval for the mean sleep duration is:

13 ± 0.1176, which rounds to (12.88, 13.12) hours with two decimal places.

Interpreting this in simple terms, we can say that we are 90% confident that the average amount of sleep for all 6-month-old infants in the U.S. is between approximately 12.88 and 13.12 hours per day. This means that based on the data collected, we can reasonably estimate that most infants in this age group sleep about 13 hours daily, with the true average likely falling within this narrow range. This level of confidence indicates a high degree of certainty, but it does not guarantee that any individual infant's sleep time falls within this interval, only that the average across many such infants would be within this estimated range.

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