• We’re interested in estimating the mean FI in the cell population.
• In experiments we cannot access the true population mean unless we measure everything.
• Sampling the reality is the next best thing we can do!

• A sample consists of measurements of FI in individual cells.
• A sample mean is the mean calculated from a sample.

• Let’s measure FI in 6 different fields of view (FOV), with 10 cells in each FOV.
• We calculate the mean FI for every FOV and obtain 6 sample means.
• The mean FI from each FOV is usually far from the population mean (=5)!

• Sample mean
  5.129893
  5.290907
  4.468820
  5.831213
  5.130247
  5.737346

• Let’s sample our biological process multiple times (e.g. by recording multiple FOVs) but at different magnifications such that we have a different number of cells per FOV.
• From each sample (=FOV) we calculate a sample mean FI.

• A sampling distribution is the distribution of a statistic, such as the mean, that is obtained by repeatedly drawing a large number of samples from a specific population.
• This distribution allows you to determine the probability of obtaining the sample statistic.

• Fortunately, we can create a plot of the distribution of sample means without collecting many different random samples!
• Instead, we’ll create a probability distribution plot using the t-distribution.

• A normal distribution describes a full population. T-distributions describe samples drawn from a full population.
• The t-distribution differs for each sample size. The larger the sample, the more it resembles a normal distribution.
• \[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \]
• Where, \(\bar{x}\) is the mean, \(s\) standard deviation, \(n\) size of the sample. \(\mu\) is the expected mean of the population.

• Let’s move to another well. This time we measured the FI of 5 cells, which yielded the mean 6.5.
• Is the new sample mean significantly different from the population mean?

• There is no magic place on the distribution to determine significant difference.
• We have a continuous decrease in the probability of obtaining sample means that are farther from the population mean.
• Where to draw the line?

This new measurement seems quite different from the population mean (=5).

• Is it still the same treatment?
• What are the chances of obtaining the sample mean (=6.5) given the population mean (=5)?
• Let’s formulate hypotheses!

• We’ve tested the null hypothesis by looking at the location of our sample mean with respect to chosen significance levels.
• How to determine the statistical significance at a chosen level without looking at the graph?
• Let’s shade the probability of obtaining our sample mean (=6.5) that is at least as extreme in both tails of the distribution (\(5 \pm 1.5\)).

• This shaded probability represents the likelihood of obtaining a sample mean that is at least as extreme as our sample mean (=6.5) in both tails of the distribution if the population mean is 5.
• That’s our P value!

• When a P value is less than or equal to the chosen significance level, we reject the null hypothesis.

• Let’s try to estimate the population mean.
• We sample once by measuring 5 cells.

• Sample
  4.273791
  5.064480
  4.419903
  3.749903
  5.279963

• Sample mean = 4.557608
• Sample SD = 0.618591

• The sample mean is the most likely value for the population mean given the information we have.
• It is a point estimate, or a best guess, of an unknown population mean.
• However, it would not be unusual to obtain different sample means if we drew other random samples from the same population.

• To put a number on the uncertainity of the point estimate, let’s shade the inner 95% of the distribution.
• 95% is an arbitrary but a commonly accepted typical range.
• A specific CI represents the margin of error, or the amount of uncertainty, around the point estimate, which is our single sample mean.
• A specific CI is a range of values that is likely to contain an unknown population parameter.
• It is NOT the probability of the population mean being in THAT range!
• The population mean is UNKNOWN but FIXED; it either is or isn’t in a CI.

• If you draw a random sample many times (from the same population!), a certain percentage of confidence intervals will contain the population mean.
• This percentage is the confidence level!
• In our case 95% of the time, the confidence interval WILL contain the population mean.

• You can use either P values or confidence intervals to determine whether your results are statistically significant.
• If a hypothesis test produces both, these results will agree.
• The confidence level is equivalent to \(1 - \alpha\).

The significance level defines the distance the sample mean must be from the null hypothesis to be considered statistically significant.

• The P-value is not a “score”.
• The P-value is NOT the probability that the null hypothesis is true.
• It is the probability of obtaining an effect at least as extreme as the one in your sample data, assuming that the null hypothesis is true.