1) determine null and alt hypo, using u where H0:u=u0 and H1 is any other one. U0 is assumed value of pop mean

2) select lvl of sig alph based on seriousness of making a type I error

3) provided that the pop from which the sample is drawn is normal or sample size is large, and pop sd σ is known , the distr of sample mean x- , u0 is normal w/ mean and sd σ=SD(x-)=σ/n

Therefore t0=(x—-u0)/s/n represents the # of sd that the sample mean is from the assumed mean. This is test statistic

4) lvl of sig used to determine the crit value, the crit region represents the mas # of sd that the sample mean can b from u0 before the null hypo is rejected. The crit region or rejection region is the set of all values such that the null hypo is rejected

Classical approach: Use table t to determine the crit value using n-1 dof

L tailed is -tα, R tail is tα, two tail is -tα/2 and tα/2

5) compare crit value w/ test statistic

Two tail: if t0< -tα/2 or t0> tα/2, reject the null hypo

L tail: if t0<-tα, reject null hypo 9like (-2.22<-1.2, so reject)

R tail: if t0>tα, reject the null hypo

6) state conclusion

P-value approach: 4) use table to determine P-value using n-1 dof

L tail: area to left of t0 is p-value, two tail: sum of area in the tail is p value (-t0 to t0), R tail: area to R of t0 is p value

Look at table, go on dof row, and find the calculated t-value or at least the two numbers its between. Go up and see what the confidence lvl is and thats the p-value

5) if p-value< α, reject the null

6) state the conclusion NOTE when dof not provided in the table, round off the df to the nearest df on the table; ex) n=78 gives df 77, round it off to 80. n=65 give df 64, round to 60. df=55, use lower closest df=50

In t-table, last row is the same as Z. this is due to the fact that t is closer to z when n is larger, therefore when n go to infinite, t is same as z, the last row on the table

However if we use computer software, we can always use t regardless the sample size, since it'll compute the most accurate t-value for us

Find t-score using eqn of mean and sd, which gives evid for either reject or accept H0 depending on hwo far from center t-distr lies mMargin of error: half of width of confidence interval ; 2xSD(pˆ) or zalpha/2• SD using sample prop to estimate pop proportion

Certainty vs precision

To b more confident, we wind up being less precise; need more values in our confidence interval to b more certain

Tension b/w certainty and precision is always there

Usually choose confidence lvl 90, 95, 99%

Generally confidence interval have form estimate +/- ME. more confident we wanna b, larger our ME needs to b, making the interval wider

Crit values

1.96 crit value z* for 95% confidence interval, 2SD only applies to 95%

Null → pi within 2 sd of obs sample prop for 95% sample

95% confidence for pi long run: pˆ+/- 2SD.

Unlike plausible values (which let sd ∆ w/ diff value pi),

Standard error SE= pˆ(1-pˆn, est of sd of statistic

In a normal distr, pˆ + z pˆ(1-pˆn ← theory based one sample Z interval. Valid if at least 10 obs unit each category. This is for population proportion pˆ, provided that npi≥10 and n(1-pi)≥10

Reaching out 2 SE on either side of pˆmake us 95% confident we'll trap true prop p