The average score for male golfers is 95 and the average...

The average score for male golfers is 95 and the average score for female golfers is 106 (Golf Digest, April $2006 ) .$ Use these values as the population means for men and women and assume that the population standard deviation is $\sigma=14$ strokes for both. A simple random sample of 30 male golfers and another simple random sample of 45 female golfers will be taken. a. Show the sampling distribution of $\overline{x}$ for male golfers. b. What is the probability that the sample mean is within three strokes of the population mean for the sample of male golfers? c. What is the probability that the sample mean is within three strokes of the population mean for the sample of female golfers? d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within three strokes of the population mean higher? Why?

The average score for male golfers is 95 and the average score for female golfers is 106 (Golf Digest, April $2006 ) .$ Use these values as the population means for men and women and assume that the population standard deviation is $\sigma=14$ strokes for both. A simple random sample of 30 male golfers and another simple random sample of 45 female golfers will be taken.
a. Show the sampling distribution of $\overline{x}$ for male golfers.
b. What is the probability that the sample mean is within three strokes of the population mean for the sample of male golfers?
c. What is the probability that the sample mean is within three strokes of the population mean for the sample of female golfers?
d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within three strokes of the population mean higher? Why?

Key Concepts

Central Limit Theorem

The Central Limit Theorem (CLT) states that, regardless of the population distribution, the distribution of the sample means will tend to be approximately normal when the sample size is sufficiently large. This principle is fundamental because it allows us to make probabilistic inferences about the sample mean using the normal distribution even when the underlying data are not normally distributed.

Sampling Distribution of the Mean

The sampling distribution of the sample mean describes how the mean of samples drawn from a population is distributed. This distribution has a mean equal to the population mean and a standard deviation, often called the standard error, equal to the population standard deviation divided by the square root of the sample size. Understanding this distribution is crucial for estimating the probability that a sample mean falls within a particular interval around the population mean.

Standard Error

The standard error is the measure of the variability of the sample mean from sample to sample. It is calculated by dividing the population standard deviation by the square root of the sample size. A smaller standard error indicates that the sample mean is likely to be closer to the population mean, which is essential for assessing precision in estimation and constructing confidence intervals.

Z-Scores and Probability Calculation

Z-scores are used to standardize a value from the sampling distribution, allowing one to compute probabilities using the standard normal distribution. By converting a range around the population mean into corresponding z-scores, one can determine the probability that the sample mean falls within that range. This process is central to hypothesis testing and other inferential statistical methods.

Effect of Sample Size on Precision

The precision of the sample mean as an estimate of the population mean is influenced by the sample size. Larger sample sizes reduce the standard error, leading to a more concentrated sampling distribution. This means that as sample size increases, the probability that the sample mean lies within a fixed distance from the population mean becomes higher. This concept underlines the importance of adequate sample sizes in achieving reliable statistical inferences.

Transcript

00:01 All right, so this question gives us the average golf score distribution for men and for women.

00:07 And it's going to ask us some questions using sampling distributions for each.

00:13 So part a wants us to graph the sampling distribution for men.

00:20 So in this case, it's just going to be a bell curve with a mean of 95 and a standard deviation of 14 over square root of the sample size.

00:55 Because we can't use the population standard deviation, because we're taking a sample of 30, so we have to correct it.

01:02 We have to use the standard error.

01:06 Then, part b asks the probability that the sample of men is within three strokes.

01:33 So that's the same thing as asking, what is the probability that the number of strokes is between 92 and 98? just, we get that from subtracting three from the mean as our lower and adding three to the mean for our upper.

02:00 And this, since we're dealing with probabilities and normal curves, we use normal cdf for, with our lower bound of 92, our upper bound of 98, our mean of 95, and our standard error, because we're dealing with a sample of 30, of the population standard deviation, divided by the square root of the sample size.

02:42 And this works out to be 0 .759.

02:56 Then it asks us the same question, except for women.

03:16 So for the women distribution, this is the same thing as asking, subtracting 3 and adding 3 from the mean.

03:27 The probability that we are between 103 and 109, which again, we use normal cdf to compute probabilities from normal curves...

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