From last lecture (Sampling Distribution):The first important bit we need to know about sampling distribution is?What is the mean of the sampling distribution of means?How do we calculate the standard deviation of the sampling distribution of means and what is it called?

Hypothesis testing:

The story so far...

The means of a sampling distribution are often normally distributed (even if the original population from which the samples came is not normal).(b) Any given sample mean ( ) can be expressed in terms of how much it differs from the mean of the sampling distribution (i.e., as a z-score).

This is true of N 30.

sample D: mean = 600gBrain size in hares: sample means and population means

sample A meanThe Central Limit Theorem in action:sample H meansample G meansample F meansample J meansample B meansample C meansample D meansample E meansample K mean, etc.....Frequency with which each sample mean occurs:the population mean, and the mean of the sample means

(c) Any given sample mean can be expressed in terms of how much it differs from the population mean.(d) "Deviation from the mean" is the same as "probability of occurrence": a sample mean which is very deviant from the population mean is unlikely to occur.

A word about the use of the term populationThe term does not necessarily refer to a set of individuals or items (e.g. cars). Rather, it refers to a state of individuals or items.

Example: After a major earthquake in a city (in which no one died) the actual set of individuals remains the same. But the anxiety level, for example, may change. The anxiety level of the individuals before and after the quake defines them as two populations.

A concrete example:An American Sludge Oil Company has developed a new petrol additive that they claim will increase the mileage of cars It is known that the current average of cars is 18 mi/gallonA sample of 100 cars is taken and run on sludge fuelThe sample mileage is 19 mi/gallonIs the Sludge companys claim valid?

When the test is fair ONLY two sources contribute to the difference between the population and the sample:

The effect of the fuel additive

Random variabilityOur job is to decide which of these possibilities is it

Hypothesis testingTwo types hypotheses: H0 and H1

The NULL hypothesis (H0): The mean of the population from which the sample comes IS 18 mi/gallon.

The ALTERNATIVE hypothesis (H1): The mean of the population from which the sample comes IS SIGNIFICANTLY DIFFERENT FROM 18 mi/gallon.

The question we want to ask is: How likely is it that the sample (N = 100) whose mean is 19 mi/gallon comes from a population whose mean and standard deviation are18 mi/gallon and 4 mi/gallon, respectively?

Assume we know the following:The population mean mileage of cars: k = 18 mi/gallonThe population standard deviation () = 4 mi/gallonThe sample mean ( ) = 19 mi/gallonNumber of cars (N) = 100

The sampling distribution for N = 100Frequencyof samplemeans = 18

We start by supposing the null hypothesis is true. Meaning that the sampling distribution from which our sample comes is one for which = 18Then we ask what is the probability that our sample came from this distribution. If the probability is low, then we reject the null hypothesis in favor of the alternative hypothesis.

Assume "innocent until proven guilty": we retain the null hypothesis unless there is enough evidence to allow us to reject it in favour of the alternative hypothesis.

Large difference between sample mean and : reject H0, and assume H1 is true.

If the difference is Small: have no reason to reject H0.

How low does the probability have to be?We usually select 0.05 as the cutoff. (Sometimes we also select 0.01). This is called the level of significance.We look for values that occur with a probability of 0.05 or less.Frequencyof samplemeans = 18

How low does the probability have to be?We usually select 0.05 as the cutoff. (Sometimes we also select 0.01). This is called the level of significance.We look for values that occur with a probability of 0.05 or less.Frequencyof samplemeans = 18

How low does the probability have to be?We usually select 0.05 as the cutoff. (Sometimes we also select 0.01). This is called the level of significance.We look for values that occur with a probability of 0.05 or less.Frequencyof samplemeans = 18

The shaded area corresponds to the 5% of the lowest and highest means ( ) in the sampling distributionFrequencyof samplemeans = 180.0250.025

Next we find the critical valuesThese are given as z-scores in the Table:Areas Under the Unit Normal DistributionFrequencyof samplemeans = 180.0250.025uppercritical value1.96 lowercritical value-1.96

z Below z Above z Betweenmean and z1.90 0.9713 0.0287 0.4713 1.91 0.9719 0.0281 0.4719 1.92 0.9725 0.0275 0.4725 1.93 0.9732 0.0268 0.4732 1.94 0.9738 0.0262 0.4738 1.95 0.9744 0.0256 0.4744 1.96 0.9750 0.0250 0.4750 1.97 0.9755 0.0245 0.4755 1.98 0.9761 0.0239 0.4761 A portion of the table:

If z-observed than 1.96then reject H0Our z-score for the sludge company is:So, we reject the Null hypothesis in favour of the alternative hypothesis

How big must a difference be, before we reject H0?

Two types errors can occur

Type 1 error: reject H0 when it is true. (Also known as , "alpha error). We think our experimental manipulation has had an effect, when in fact it has not.

Type 2 error: retain H0 when it is false. (Also known as , "beta error). We think our experimental manipulation has not had an effect, when in fact it has.

Any observed difference between two sample means could in principle be either "real" or due to chance - we can never tell for certain.

But -Large differences between samples from the same population are unlikely to arise by chance.

Small differences between samples are likely to have arisen by chance.

Problem: reducing the chances of making a Type 1 error increases the chances of making a Type 2 error, and vice versa.

Psychologists therefore compromise between the chances of making a Type 1 error, and the chances of making a Type 2 error:

We set the probability of making a Type 1 error at 0.05. When we do an experiment, we accept a difference between two samples as "real", if a difference of that size would be likely to occur, by chance, 5% of the time, i.e. five times in every hundred experiments performed.

Directional and non-directional hypotheses:

Non-directional (two-tailed) hypothesis:Merely predicts that sample mean ( ) will be significantly different from population mean ()> : Differences this extreme (or more) occur by chance with p = 0.025. = > < < : Differences this extreme (or more) occur by chance with p = 0.025.possible differences

Directional (one-tailed) hypothesis:More precise - predicts the direction of difference (i.e., either predicts is bigger than , or predicts is smaller than ). > : Differences this extreme (or more) occur by chance with p = 0.05.possible differences

*SAMPLE SIZE

SAME AS THE MEAN OF THE POPULATION

-POPULATION S.D. DIVIDED BY THE SQ. ROOT OF N-STANDARD ERROR*Drawing conclusions about populationsbased on observations of samples.

Hypothesis testing Can be very powerful.It can lead to understanding of cause and effect.Does smoking cause lung cancer?

**Many factors could contribute: The sample had Better cars; better drivers; better driving conditions; more air in the tyres

*When we pick a sample we immediately start thinking about the sampling distribution fromwhich the sample came. This is of course a theoretical distribution.*What is the mean of this distribution?Can we find out what the standard deviation of this distribution is, given the population S.D.= 4?And what is the standard deviation of such a distribution called?Does this mean that all the means in this distribution will be 18? NO!*Can we find out what the standard deviation of this distribution is?And what is it called?*The distribution contains ALL the sample means. Many of the means are around 18 mi/gallon. But few may beas high as 20 or as low as 16. We would to keep only 5% of all means from the highest and the lowest groups.*We throw out many X-Bar values that occur most frequently and ask: are there 5% of x-bar values left?The answer is NO!*So we throw out some more X-Bar values (which still occur quite frequently). We keep goinguntil we have left 5% of the least frequently occurring x-bar values.*We stop throwing out x-bar values until the two shaded areas(TOGETHER) make up 5% of the total area of the distribution.*We stop throwing out x-bar values until the two shaded areas(TOGETHER) make up 5% of the total area of the distribution.