CHAPTER 14

• Chapter 14 introduces statistical inference

Statistical Estimation

• Different samples from a population yield different sample means
• Virtually none of the sample means will equal the true population mean.
• Sampling distribution is N(μ, σ/√n) if the population is N(μ, σ)
• Sampling distribution is ~ N(μ, σ/√n) if n is large enough.
• One random sample of size n is taken to rely on the known properties of the sampling distribution.

Visualization of Confidence

• Random samples taken using a computer applet allow calculation of the sample mean and an interval
• The interval size is plus-or-minus 2 σ/√n around the mean
• Approximately 95% of computed intervals capture the parameter μ, based on the 68-95-99.7% rule.
• The confidence interval is a range of values with an associated probability, also called the confidence level, C
• The probability quantifies the likelihood that the interval contains the unknown population parameter
• Confidence C indicates the probability that μ falls within the computed interval

Margin of Error

• A confidence interval ("CI") can be expressed as a center ± a margin of error m, resulting in μ within m.
• It can also be expressed as an interval, with μ falling within a range from (m) to (m)
• The confidence level C (in %) indicates the area of corresponding size C under the sampling distribution.
• A 95% confidence interval for mean body temperature (°F) computed to be (98.1, 98.4) from a sample of 130 healthy adults means: There is 95% confidence that the population mean body temperature is a value between 98.1 and 98.4°F
• For the same 95% confidence interval (98.1, 98.4), the margin of error is 0.15°F.

Confidence Interval

• Confidence interval, C, when taking a random sample from a Normal population with known standard deviation σ, the level C confidence interval for μ is: σ/√n
• σ/√n is the standard deviation of the sampling distribution
• C is the area under the N(0,1) between -z* and z*
• Z- and t-values (Table C) can be used
• The appropriate z*-value for a confidence level C is listed in the same column.

Confidence Interval Calculations

• Measurement equipment has Normal distribution with standard deviation σ = 1 million bacteria/mL of fluid
• Three measurements made: 24, 29, and 31 million bacteria/mL
• Mean: 28 million bacteria/ml
• Find the 99% and 90% CI.
• 99% confidence interval for the true density, z* = 2. 576

Confidence Level and Margin of Error

• Confidence level C determines the value of z* (in a table)
• The margin of error depends on z*.
• Higher confidence C implies a larger margin of error m (less precision more accuracy)
• Lower confidence level C yields a smaller margin of error m (more precision less accuracy)
• High confidence and narrow intervals are desired
• Choose the confidence level first, then increase n to achieve a narrow interval.

Hypothesis Testing

• A claim about the unknown value of a population parameter is made
• An analysis is performed to check if this claim is reasonable based on the evidence gathered (sample data).
• A test of statistical significance is performed to tests a specific hypothesis
• The sample data is used to decide on the validity of the tested hypothesis.
• Inorganic phosphorus blood levels vary Normally among adults, with mean 1.2 and standard deviation 0.1 mmol/L
• Average inorganic phosphorus level of a random sample of 12 healthy elderly subjects is 1.128 mmol/L
• Question to be asked: Is this smaller mean phosphorus level due to chance variation alone, or is it evidence that the true mean phosphorus level in elderly individuals is lower than 1.2 mmol/L?

Null and Alternative Hypothesis

• Null hypothesis, Ho, is a specific statement about a parameter of the studied population(s)
• Alternative hypothesis, H₂, is a more general statement that complements and is mutually exclusive with the null hypothesis.
• Phosphorus levels Example:
• H₀:= μ = 1.2 mmol/L
• H₂: μ < 1.2 mmol/L (μ is smaller due to changing physiology)

One-Sided versus Two-Sided Alternatives

• A two-tail (two-sided) alternative is symmetric: H₂: μ [value]
• A one-tail (one-sided) alternative is asymmetric and specific:
• H₂: μ < [value]
• H₂: μ > [value]
• Choice of a one-sided versus two-sided test depends on the question being asked and prior knowledge about the problem.
• If asymmetric, H₂ should be one-sided, otherwise, H₂ should be two-sided.
• Examples of hypothesis:
• Is active ingredient concentration as stated on the label (325 mg/tablet)?
• H₀: μ = 325 vs H₂: μ ≠ 325
• Is Nicotine content > written value of 1mg/cigarette on average?
• H₀: μ = 1 vs H₂: μ > 1
• Does a Drug cause a change in Blood Pressure on average?
• H₀: μ = 0 vs H₂: μ ≠ 0
• Does a particular stream have an unhealthy mean Oxygen content (a level below 5mg/liter)? Ecologists collect a liter of water from each of 45 locations along and find a mean of 4.62
• H₀:=5 vs H₂:<5

P-Value and Hypothesis Testing

• Phosphorus levels vary Normally with standard deviation s = 0.1 mmol/L
• H₀: μ = 1.2 mmol/L versus H₂: μ < 1.2 mmol/L
• Mean phosphorus level from 12 elderly subjects is 1.128 mmol/L
• Evaluate probability of drawing a random sample with a mean as small as this one (or even smaller), if Ho is true
• P-value: Probability (Ho was true) of obtaining a sample statistic at least as extreme (in the direction of H₂) as the one obtained.
• Visualizing the P-value can also be done
• The chance of seeing a sample mean less than our observed average of 1.128 is 0.0063.

Interpreting the P-Value

• Check for amount of random variation that alone accounts for the difference between Ho observations from a random sample
• Small P-values are strong evidence AGAINST Ho
• In that case we reject Ho, and the findings are statistically significant
• P-values that are not small fail to give enough evidence against Ho
• There is a failure to reject Ho and Ho cannot be proved

Range of P-value

• P-values are probabilities and therefore always between 0 and 1.
• The order of magnitude of the P-value matters more than its exact numerical value.
• Significance level, α, is the largest P-value is tolerated for rejecting Ho
• This value is decided arbitrarily before conducting the test.
• When P-value ≤ α, reject Ho
• When P-value > α, it leads to a failure to reject Ho
• Example:
  • If industry standards require significance level α of 5% and the two-sided is performed on a sample of data where
  • P-value is 4.56%, then the results are statistically significant at significance level 0.05.
• To test H₀: μ = μ₀ using a random sample of size n from a Normal population with known standard deviation σ, usethe null sampling distribution N(μ₀, σ/√n).
• P-value represents the area under N(μ₀, σ/√n) for values of x at least as extreme in the direction of H₂ as that of a random sample.
• Calculate the z-value then consult Table B or C or use appropriate technology

P-Value Symmetry

• Calculate the P-value for a two-sided test using the symmetry of the normal curve.
• The P-value for a one-sided test is found and doubled for asymmetry
• Examples on slides 33 and 34 test the hypothesis:
• Are there statistically significant differences blood phosphorus levels between healthy adults (1.2mmol/L) and elderly adults (1.128mmol/L)?
• P-value=0.0063
• This shows the mean phosphorus level among the elderly is significantly less than 1.2 mmol/L

Testing Confidence Intervals

• Because a two-sided test is symmetric, a confidence interval can easily test a two-sided hypothesis
• In a two-sided test, C = 1 − α, where C is the confidence level and α is the significance level.

Logic of Confidence Interval

• A 99% CI found for the true bacterial density is 26.5 to 29.5 million bacteria/mL.
• With 99% confidence, could the population mean be μ = 25 million/mL or μ = 29?
• A confidence interval offers a yes/no answer and an estimated range of likely estimates the range for the true population mean μ.
• A P-value quantifies how strong the evidence is against H₀
• But if H₀ is rejected, it provides no information about the true population mean μ.
• This section discussed the concept of statistical inference, covering statistical estimation, margin of error, confidence levels, confidence intervals, hypothesis testing, P-values, and tests for a population mean