Normal probability distributions part III and introduction to estimation

Courtesy of Melikamp CC via Wikimedia Commons

• Recall now the bell curve picture that we often consider – we will suppose we have a population that is distributed as a bell shape.
• We suppose that the population mean is \( \mu \) and population standard deviation \( \sigma \).
• Normally distributed data is characterized by the following features:
1. The frequencies start low, then increase to one or two high frequencies, and then decrease to a low frequency.
2. The distribution is approximately symmetric.
3. There are few if any extreme values.
• These features can be understood as a direct application of the empirical rule.
• We suppose that the histogram represents the sample data which is mostly bell-shaped, but the collection is smaller than the population so it is not exact.
• In particular, any data set is subject to sampling error and we cannot expect a perfect bell shape from a small sample even when the population is perfectly normally distributed.

• So far, we have used histograms to examine these features in data and to assess normality.
• However, histograms sometimes have technical issues that make them unreliable.

• There is no “perfect” choice generally for how wide the bins should be in a histogram, and it makes this less reliable for assessing normality than other methods.
• For this reason, we will introduce a more advanced visual method for assessing normality.
• If we want to examine if data is “close-enough-to-normal”, it makes sense to compare our data to the normal distribution directly;
• to do so, we will introduce the Q-Q or quantile-quantile plot.

Courtesy of Glen_b via the Stack Exchange

• We will discuss formaly how Q-Q plots are constructed as follows.
• Note: we will discuss the version of Q-Q plots described in the book for simplicity, but typically the Q-Q plots used in practice have a slightly different construction.
• Suppose we have some collection of sample data \( \{x_i\} \) for \( i=1,\cdots, n \).
• We will order our observations, \( x_1, x_2, \cdots x_n \) (changing the index if necessary) such that, \[ x_i \leq x_{i+1} \] for every \( i=1,\cdots, n-1 \).
• On the horizontal axis of the Q-Q plot, we use the scale of the original data.
• On the vertical axis, we use the scale of z scores.

• Combined, the Q-Q plot places the z-score of each observation above the raw value in the horizontal axis.
• If the sample data is “perfectly” normal, then the z scores should form a straight line against the ordered data.
• However, sampling error will lead to small deviations from this straight line in practice.

• For the IQ data histogram earlier, we see the associated Q-Q plot to its right.
• Here, this is mostly a straight line with only slight deviations in the tails.
• However, the data doesn’t have many extreme values, so it isn’t a concern.

• What we should detect in this Q-Q plot is the following:
1. The distribution is symmetric, as can be seen by the symmetry in the Q-Q plot.
2. However, we can see a smaller relative concentration of values towards the mean as evidenced by the “bowing” of the curve above and below the line.
• Near the interior, the magnitude of the z scores should be increasing faster based on the percentiles of the observed data.
• At the tails, there are very extreme values that we would not expect to see in normal data.

Courtesy of Pearson

• Constructing Q-Q plots manually is tedious and should be done with computer software.
• For this reason, we will only consider how to construct and analyze Q-Q plots in StatCrunch.
• NOTE: to construct the Q-Q plot as in the book, you must select “Normal quantiles on y-axis” as highlighted in the figure to the left.
• We will go through some examples in the video on how to open a book data set and analyze this data set with a histogram and with a Q-Q plot.
• In the midterm, you will be expected to be able to analyze data in StatCrunch as in the following.
• You will likely be asked to produce plots of one of the variables and analyze the shape for signs of:
1. normality;
2. skewness; or
3. uniformity
• in the distribution.
• You may use any combination of Q-Q plots and histograms, and we will consider both in the following.

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

• In the last lecture, we saw how a sample proportion generates a random variable.
• That is, when we take a sample of a population and compute the proportion of the sample for which some statement is true.
• Suppose we want to replicate this sampling procedure infinitely many times…
• It impossible to replicate the sampling infinitely many times, but we can construct a probabilistic model for this replication process with a probability distribution.

• Formally, we will define \( \hat{p} \) to be the random variable equal to the proportion derived from a random sample of \( n \) observations.
• For each replication, \( \hat{p} \) attains a different value based on chance.
• Then, for random, independent samples, \( \hat{p} \) tends to be normally distributed about \( p \).
• We can thus use the value of \( \hat{p} \) and the distribution of \( \hat{p} \) to estimate \( p \) and how close we are to it.
• We know that \( \hat{p} \) is an unbiased estimator of the true population proportion \( p \).
• That is, over infinitely many resamplings, the expected value (mean of the probability distribution) for \( \hat{p} \) is equal to \( p \).
• When we have a specific sample data set, and a specific value for \( \hat{p} \) associated to it, \( \hat{p} \) is called a point estimate for \( p \).
• The measure of “how-close” we think this is to the true value is called a confidence interval.

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

• In addition to the review on the last slide we will briefly discuss why \( \hat{p} \) is a “best” estimate for \( p \)
• Recall, \( \hat{p} \) as a random variable tends to have a normal distribution around \( p \).
• This distribution can be characterized in the same way as any other normal distribution, in terms of:
1. its mean \( \mu_\hat{p} = p \); and
2. its standard deviation \( \sigma_\hat{p} \).

• Let’s suppose that \( \tilde{p} \) is some other estimate for \( p \) that is unbiased, i.e., \[ \mu_\tilde{p} = p. \]
• It is an extremely important property that the standard deviation of the other estimate is at least as large as \( \sigma_\hat{p} \), i.e., \[ \sigma_\tilde{p} \geq \sigma_\hat{p}. \]
• In plain English this says that

Even though \( \hat{p} \) tends to vary around \( p \) due to sampling error, the ammount it varies away from \( p \) tends to be less than all other unbiased estimators.

• We can think of the sample proportion as the most accurate point estimate, if we were to replicate samples arbitrarily many times.
• This is fortunate because it is also the most “natural” choice for an estimate in some sense.

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

• As a schematic of this, consider the figure to the left.
• In the figure, each vertical line will represent a different replication of the sampling proceedure.
• We will suppose that the true population proportion is actually \( p=0.50 \).
• Notice that our Gallup poll sample confidence interval, \[ 0.405 < {\color{#1b9e77} p } < 0.455, \] does not actually contain the true population parameter – this is due to sampling error.
• However, if we take enough replications of the sampling procedure, \( p=0.50 \) should lie in the associated confidence interval about \( 95\% \) of the time.

• Remember, in this “frequentist” statistical framework, \( p \) is not random, it is a fixed but unkown value.
• On the other hand, our confidence intervals are random and depend on a particular outcome of the sampling replication.
• Note: when we have a particular sample data set in hand, \( \hat{p} \) is also just a fixed point estimate, and its confidence interval will also be fixed.
• Therefore, we say that the \( 95\% \) confidence interval represents a procedure that should work \( 95\% \) of the time;
• we do not guarantee, however, that \( p \) actually lies within this range.

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

• We will now discuss confidence intervals (CI) more formally.
• Suppose that we want to estimate the true proportion \( p \) with some level of confidence:
• if we replicated the sampling procedure infinitely many times, the average number of times we found \( p \) in our confidence interval would be equal to the level of confidence.

• Let’s take an example confidence level of \( 95\% \) – this corresponds to a rate of failure of \( 5\% \) over infinitely many replications.
• Generally, we will write the confidence level as, \[ (1 - \alpha) \times 100\% \] so that we can associate this confidence level with its rate of failure \( \alpha \).
• Recall, we earlier studied ways that we can compute the critical value associated to some \( \alpha \) for the normal distribution.
• We will use the same principle here to find how wide is the interval around \( p \) for which \( \hat{p} \) will lie in this interval \( (1-\alpha)\times 100\% \) of the time.

• we want to find the critical value \( z_\frac{\alpha}{2} \) for which:
• \( (1-\frac{\alpha}{2})\times 100\% \) of the area under the normal density lies to the left of \( z_\frac{\alpha}{2} \); and
• \( (1-\frac{\alpha}{2})\times 100\% \) of the area under the normal density lies to the right of \( -z_\frac{\alpha}{2} \).
• Put together, \( (1-\alpha)\times 100\% \) of values lie within \( [-z_\frac{\alpha}{2},z_\frac{\alpha}{2}] \).

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

• In the figure to the left, we see exactly thus how to find the width of the region for a given confidence level.
• For a given confidence level \( (1 -\alpha)\times 100\% \), we will find the particular \( \alpha \).
• We then find the associated two-sided measure of extremeness with the \( z_\frac{\alpha}{2} \) critical value.
• This critical value is associated to the measure of extremeness of finding an observation that lies far away from the mean.
• Particularly, only \( \alpha \times 100\% \) of the population lies outside of the region \( [-z_\frac{\alpha}{2},z_\frac{\alpha}{2}] \).

• Put another way, if you randomly draw a population member, there is \( (1 -\alpha)\times 100\% \) chance that the mean lies at a distance at most \( z_\frac{\alpha}{2} \) away from this observation.
• The sample proportion point estimate \( \hat{p} \) can be considered a random draw from population of all point estimates distributed around \( p \).
• We do not know how close a particular point estimate (depending on a particular sample) is to \( p \).
• However, we know that over all possible replications of the sampling procedure, the point estimate \( \hat{p} \) will lie at a distance at most \( z_\frac{\alpha}{2} \) exactly \( (1-\alpha)\times 100\% \) of the time.
• Therefore, we have \( (1-\alpha)\times 100\% \) confidence that the true \( p \) lies within this region about \( \hat{p} \).