Confidence Intervals for Means - Notes

Chapter 11: Confidence Intervals for Means

11.1 The Central Limit Theorem

• When sampling at random, sample proportions vary; the Normal model effectively summarizes this variation. Understanding this variability is crucial in statistical inference.

• Means also possess a sampling distribution that can be modeled using a Normal model. This allows us to make inferences about population means based on sample means.

• Simulating the Sampling Distribution of a Mean illustrates how the distribution changes with the number of dice tossed:

  • One fair die: Uniform distribution. Each outcome (1 to 6) is equally likely.

  • Two fair dice: Triangular distribution. The sum of the two dice tends towards the middle values.

  • Five dice: Distribution becomes bell-shaped, approaching the Normal model. The sum is more concentrated around the mean.

  • Twenty dice: Normal shape with a smaller spread. The more dice, the closer to a perfect Normal distribution, and the less variability.

• Central Limit Theorem (CLT): The sampling distribution of any mean becomes Normal as the sample size grows, regardless of the population distribution's shape. This is a cornerstone of statistical inference, enabling us to use Normal models even when the population is not normally distributed.

  • For very skewed population distributions, a large sample size (dozens or hundreds) may be needed for the Normal model to work well. The more skewed the data, the larger the sample size needed to ensure the sampling distribution of the mean is approximately Normal.

• Two distributions to consider:

  • The real-world distribution of the sample. This is the actual data you collect.

  • The math-world sampling distribution of the statistic. This is a theoretical distribution describing how the statistic (e.g., sample mean) varies across different samples.

• The CLT concerns the sample means and proportions of different random samples from the same population, not the distribution of data from a single sample. It's about how the means of many samples behave, not about individual data points.

• It is more surprising to find one person over 6'9" tall than to find the mean height of 100 students over 6'9", because means have smaller standard deviations than individual values. This highlights how averaging reduces variability.

11.2 The Sampling Distribution of the Mean

• The Normal model for the sampling distribution of the mean has a standard deviation equal to $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation. This formula shows that as the sample size increases, the standard deviation of the sampling distribution decreases.

• The standard deviation parameter of the sampling distribution model for the sample mean $\bar{y}$ is written as $\sigma(\bar{y}) = SD(\bar{y}) = \frac{\sigma}{\sqrt{n}}$. This notation clarifies that we are referring to the standard deviation of the sampling distribution of the sample mean.

• Sampling Distribution Model for a Mean:

  • When a random sample is drawn from any population with mean $\mu$ and standard deviation $\sigma$, its sample mean $\bar{y}$ has a sampling distribution with the same mean $\mu$ but with a standard deviation of $\\frac{\sigma}{\sqrt{n}}$. This is a formal statement of how the sampling distribution behaves.

  • The shape of the sampling distribution is approximately Normal if the sample size is large enough, regardless of the population. Larger samples result in a closer approximation to the Normal distribution. The larger the sample, the more closely the sampling distribution approximates a Normal distribution, regardless of the shape of the original population distribution.

• Two related sampling distribution models:

  • Categorical data: Sample proportion $\hat{p}$ follows a Normal model with mean $p$ and standard deviation $\sqrt{\frac{p(1-p)}{n}}$. For categorical data, we use sample proportions and a slightly different formula for the standard deviation.

  • Quantitative data: Sample mean $\bar{y}$ follows a Normal model with mean $\\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$. For quantitative data, we use sample means, and the standard deviation depends on the population standard deviation and the sample size.

• Assumptions and Conditions for the Sampling Distribution of the Mean:

  • Independence Assumption: Sampled values must be independent. Each data point should not influence others.

  • Randomization Condition: Data values must be sampled randomly. Random sampling ensures that the sample is representative of the population.

  • Sample Size Assumption: The sample size must be sufficiently large.

    • 10% Condition: When sampling without replacement, $n$ should be no more than 10% of the population. This ensures independence when sampling without replacement.

    • Large Enough Sample Condition: If the population is unimodal and symmetric, a small sample is sufficient. Skewed distributions may require samples of several hundred. Always plot the data to check. A histogram or other visual display can help assess the shape of the distribution.

• Example: Cholesterol levels in healthy U.S. adults average 215 mg/dL with a standard deviation of 30 mg/dL and are roughly symmetric and unimodal. A random sample of 42 healthy U.S. adults is taken.

  • Conditions met to use the normal model:

    • Randomization: The sample is random. This ensures the sample is representative of the population.

    • 10% Condition: 42 healthy U.S. adults are less than 10% of the population of healthy U.S. adults. This condition is met because the sample size is small compared to the population.

    • Large Enough Sample Condition: Cholesterol levels are roughly symmetric and unimodal, so a sample size of 42 is sufficient. The distribution is roughly symmetric and unimodal, so the sample size is sufficient to use the Normal model.

• Mean of the sampling distribution: $\mu = 215$ mg/dL. The mean of the sampling distribution is the same as the population mean.

• Standard deviation of the sampling distribution: $\sigma_{\bar{y}} = \frac{\sigma}{\sqrt{n}} = \frac{30}{\sqrt{42}} \approx 4.629$ mg/dL. This calculation shows how the standard deviation of the sampling distribution is affected by the sample size.

• Probability of average cholesterol level being greater than 220 mg/dL: calculated using the Normal model with mean 215 and standard deviation 4.629. This is a typical application of the sampling distribution to calculate probabilities.

• Example: Mean weight of boxes shipped by a company is 12 lbs with a standard deviation of 4 lbs. Boxes are shipped in palettes of 10, with a limit of 150 lbs per shipment. What's the probability a palette will exceed the limit?

  • Probability that the total weight of 10 boxes exceeds 150 lbs is the same as the probability that the mean weight exceeds 15 lbs. This simplifies the problem by focusing on the mean weight rather than the total weight.

  • Conditions:

    • Assume 10 boxes are a random sample and weights are mutually independent. This assumption is necessary for the sampling distribution to be valid.

    • 10 boxes are less than 10% of the population of boxes shipped. This condition helps ensure independence.

  • Sampling distribution is Normal with mean 12 and standard deviation $\frac{4}{\sqrt{10}} \approx 1.265$. This describes the distribution of sample means.

  • P(mean > 15) = 0.0087 (less than 1%). This is the probability that a palette will exceed the weight limit.

11.3 How Sampling Distribution Models Work

• Standard Error (SE): Estimate of the standard deviation of a sampling distribution. Since we often don't know the true population standard deviation, we estimate it using the standard error.

  • For a sample proportion, $\hat{p}$, the standard error is: $SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

  • For the sample mean, $\bar{y}$, the standard error is: $SE(\bar{y}) = \frac{s}{\sqrt{n}}$.

• The sample proportion and sample mean are random quantities affected by sample variation. Understanding that these statistics vary from sample to sample is fundamental to statistical inference.

• Two truths about sampling distributions:

  • Sampling distributions arise because samples vary. If every sample was identical, there would be no need for sampling distributions.

  • The Central Limit Theorem simplifies the process of simulating sampling distributions for means and proportions. The CLT allows us to use Normal models, making calculations much easier.

• Diagram of relationships:

  • Start with a population model (mean $\mu$, standard deviation $\\sigma$). This is the true, but often unknown, distribution of the population.

  • Draw one real sample (size $n$, histogram, summary statistics). This is the data we collect and use to make inferences.

  • Imagine many other samples. This is a thought experiment to understand how sample statistics vary.

  • Gather all means into a histogram, which can be modeled with a Normal model (mean $\mu$, standard deviation $\frac{\sigma}{\sqrt{n}}$). The CLT tells us that this histogram will be approximately Normal.

  • Estimate $\sigma$ with the sample standard deviation $s$ to get the standard error: $SE = \frac{s}{\sqrt{n}}$. Since we usually don't know $\sigma$, we use $s$ to estimate it.

11.4 Gosset and the t-Distribution

• Confidence intervals for proportions: $\hat{p} \pm z^* \times SE(\hat{p})$, where ME = critical value * SE($\hat{p}$). This is the formula for a confidence interval for a population proportion.

• Confidence intervals for means: $\bar{y} \pm t^* \times SE(\bar{y})$, where ME = critical value * SE($\bar{y}$). This is the formula for a confidence interval for a population mean.

• Standard deviation of the sample mean: Need to know true value of population standard deviation $\sigma$. Ideally, we would use the population standard deviation, but this is often unknown.

• Use $s$, the sample standard deviation, instead of $\sigma$ to compute standard error. Since we often don't know $\sigma$, we use $s$ as an estimate.

• William S. Gosset discovered that using the standard error $SE = \frac{s}{\sqrt{n}}$ resulted in a non-Normal curve. This discovery led to the development of the t-distribution.

• He developed the Student's t-model: bell-shaped, but details change with sample sizes. The t-distribution accounts for the extra uncertainty introduced when using $s$ instead of $\sigma$.

• Student’s t-models form a family of related distributions depending on degrees of freedom. The degrees of freedom determine the shape of the t-distribution.

• t-models:

  • Unimodal, symmetric, and bell-shaped (like the Normal model). The t-distribution is similar in shape to the Normal distribution.

  • t-models with few degrees of freedom have narrower peaks and fatter tails than the Normal model. This reflects the increased uncertainty with smaller sample sizes.

  • As degrees of freedom increase, t-models approach the Normal model. With larger sample sizes, the t-distribution becomes very similar to the Normal distribution.

• Example: Survey of 25 customers found a mean age of 31.84 years and a standard deviation of 9.84 years.

  • Standard error of the mean: $SE(\bar{y}) = \frac{9.84}{\sqrt{25}} = 1.968$ years. This is a calculation of the standard error using the sample standard deviation.

  • If the sample size was 100 instead of 25: $SE(\bar{y}) = \frac{9.84}{\sqrt{100}} = 0.984$ years. This demonstrates how increasing the sample size reduces the standard error.

• Practical Sampling Distribution Model for Means: When certain conditions are met, the standardized sample mean,
  

$\qquad t = \frac{\bar{y} - \mu}{SE(\bar{y})}$

follows a Student's t-model with $n-1$ degrees of freedom. The standard error can be found from:

$\qquad SE(\bar{y}) = \frac{s}{\sqrt{n}}$.

11.5 A Confidence Interval for Means

• One-Sample t-Interval: When assumptions and conditions are met, the one-sample t-interval for the population mean $\mu$ is: $\bar{y} \pm t_{n-1}^* \times SE(\bar{y})$

  • The standard error of the mean is: $SE(\bar{y}) = \frac{s}{\sqrt{n}}$.

  • The critical value $t_{n-1}^*$ depends on the confidence level $C$ and degrees of freedom $n-1$. The critical value is obtained from a t-table or software and depends on the desired confidence level and the degrees of freedom.

• Degrees of Freedom – Why n – 1?

  • If the true population mean $\mu$ is known, the standard deviation can be found using $n$. When we know the population mean, we can calculate the standard deviation directly.

  • Since we use $\bar{y}$ instead of $\mu$, $\bar{y}$ is as close to the data values as possible, making $s$ too small. Using the sample mean makes the sample standard deviation an underestimate of the population standard deviation.

  • Compensate by dividing by $n – 1$ instead of by $n$. Dividing by $n-1$ instead of $n$ provides a better estimate of the population standard deviation.

• Finding t-Values:

  • The Student’s t-model varies for each value of degrees of freedom. Each degrees of freedom corresponds to a unique t-distribution.

  • t-tables provide critical values for common confidence levels (80%, 90%, 95%, 99%). t-tables are a convenient way to look up critical values.

  • Technology can provide critical values for any number of degrees of freedom and confidence level. Software packages can calculate precise critical values.

  • As df increases, t-models get closer to the Normal model; the final row in a t-table has critical values from the Normal model and is labeled “∞”. As the degrees of freedom increase, the t-distribution approaches the Normal distribution.

• Example: Survey of 25 customers, mean age 31.84 years, standard deviation 9.84 years. Construct a 95% confidence interval for the mean.

  • $SE(\bar{y}) = \frac{9.84}{\sqrt{25}} = 1.968$

  • df = 25 - 1 = 24

  • $t_{24}^*$ for 95% confidence = 2.064

  • 95% CI: 31.84 ± 2.064 * 1.968 = (27.78, 35.90).

  • Interpretation: We’re 95% confident the true mean age of all customers is between 27.78 and 35.90 years. This is the correct interpretation of a confidence interval.

11.6 Assumptions and Conditions

• Independence Assumption: Cannot be directly checked; consider if the assumption is reasonable. We must decide whether it is reasonable to assume the data values are independent.

• Randomization Condition: Data from a random sample or randomized experiment. Random sampling or a randomized experiment is essential for valid inference.

• 10% Condition: Sample size should be no more than 10% of the population (typically met for means). This condition helps to ensure independence when sampling without replacement.

• Normal Population Assumption: t-models do not work for badly skewed data; assume data come from a Normal model. The t-models are most reliable when the data come from a Normal model, but they are robust to moderate departures from Normality.

• Nearly Normal Condition: Data come from a unimodal and symmetric distribution (check with a histogram).

  • For very small samples (n < 15), data should closely follow a Normal model; t methods are not appropriate if there are outliers or strong skewness. With very small samples, the data must be very close to Normal.

  • For moderate sample sizes (n between 15 and 40), t methods work well if data are unimodal and reasonably symmetric. With moderate sample sizes, the t methods are quite robust.

  • For larger sample sizes (n > 40 or 50), t methods are safe unless data are extremely skewed; analyze with and without outliers if present. With larger sample sizes, the t methods are generally safe unless the data are extremely skewed or have significant outliers.

• Example: Compensation of 500 CEOs shows an extremely skewed distribution. In this case, t-methods might not be appropriate without careful consideration.

• Taking many samples of 100 CEOs results in a nearly Normal plot for the sample means. This illustrates the Central Limit Theorem in action.

• Check Assumptions and Conditions; Example: Survey of 25 customers, mean age 31.84 years, standard deviation 9.84 years, 95% CI (27.78, 35.90).

  • Independence: Data were gathered from a random sample and should be independent. This is a judgment based on how the data were collected.

  • 10% Condition: These customers are fewer than 10% of the customer population. This condition is easily met in this case.

  • Nearly Normal: The histogram is unimodal and approximately symmetric. We need to examine a histogram or other display of the data to assess this condition.

• Cautions about Interpreting Confidence Intervals

  • The confidence interval is about the mean, not individual observations. Confidence intervals estimate the population mean, not individual data values.

  • Your uncertainty is about the interval, not the true mean. The interval varies randomly. The true mean is fixed, but the confidence interval varies from sample to sample.

  • Make sure to say that you’re 95% confident that your interval contains the true mean. This is the correct way to express confidence in the interval.

What Can Go Wrong?

• Decide when to use Student’s t methods.

  • Don’t confuse proportions and means.

    • Use Normal models with proportions. When working with categorical data, use Normal models for proportions.

    • Use Student’s t methods with means. When working with quantitative data, use Student’s t methods for means.

  • Beware of multimodality. If you see this, try to separate the data into groups. Multimodal data may indicate that there are distinct subgroups in the data.

  • Beware of skewed data. If it is skewed, try re-expressing the data. Skewed data can violate the assumptions of t-methods; consider transformations like logarithms.

• Investigate outliers.

  • If they are clearly in error, remove them. Outliers that are due to data entry errors should be corrected or removed.

  • If they can’t be removed, you might run the analysis with and without the outlier. Analyze the data both with and without outliers to assess their impact.

• Watch out for bias. Measurements can be biased. Biased measurements will lead to incorrect conclusions.

• Make sure data are independent. Consider whether there are likely violations of independence in the data collection methods. Violations of independence can invalidate the results of the analysis.

From Learning to Earning

• Know the sampling distribution of the mean.

  • To apply the Central Limit Theorem for the mean in practical applications, we must estimate the standard deviation. This standard error is
    

    $\qquad SE = \frac{s}{\sqrt{n}}$.\

  • When we use the SE, the sampling distribution that allows for the additional uncertainty is Student’s t. The t-distribution accounts for the extra uncertainty when we estimate the standard deviation.

• Construct confidence intervals for the true mean, $\mu$.

  • A confidence interval for the mean has the form
    

    $\qquad \bar{y} \pm t^*SE$

  • The Margin of Error is
    

    $\qquad ME = t^*SE$

• Find t* values by technology or from tables. Use t-tables or software to find the appropriate critical value.

• When constructing confidence intervals for means, the correct degrees of freedom is n – 1. Always use n-1 degrees of freedom for a one-sample t-interval.

• Check the Assumptions and Conditions before using any sampling distribution for inference. Always verify that the assumptions and conditions are met before proceeding with the analysis.

• Write clear summaries to interpret a confidence interval. Clearly explain what the confidence interval means in the context of the problem.

• Be able to perform a hypothesis test for a mean.

  • The null hypothesis has the form
    

    $\qquad H_0: \mu = value$

• We refer the test statistic to the Student’s t distribution with n – 1 degrees of freedom. Use the t-distribution with n-1 degrees of freedom to calculate the p-value.