A study measured how fast subjects could repeated push a button when under the effects of the caffeine in Mountain Dew soda. Subjects were asked to push a button as many times as possible in two minutes after consuming the soda. During another test session they were asked to push the button after drinking a placebo. The subjects did not know which treatment they were administered each day and the order of the treatments was randomly assigned. The data, given in presses per two minutes for each treatment is on the right. Is there evidence to determine whether or not the caffeine in Mountain Dew results in a higher rate of beats, on average, per two-minute period? Perform a hypothesis test at the 0.10 level of significance and state your conclusion. Beats Caffeine 281 284 300 421 240 294 377 345 303 340 408Beats Placebo 201 262 283 290 259 291 354 346 283 391 411

In a sample of 2023 u. s. adults, 373 said franklin roosevelt was the best president since world war ii. two u. s. adults are selected at random from the population of all u. s. adults without replacement. assuming the sample is representative of all u. s. adults, complete parts (a) through (d). (a) find the probability that both adults say franklin roosevelt was the best president since world war ii. the probability that both adults say franklin roosevelt was the best president since world war ii is (round to three decimal places as needed.) (b) find the probability that neither adult says franklin roosevelt was the best president since world war ii. the probability that neither adult says franklin roosevelt was the best president since world war ii is (round to three decimal places as needed.) (c) find the probability that at least one of the two adults says franklin roosevelt was the best president since world war ii. the probability that at least one of the two adults says franklin roosevelt was the best president since world war ii is (round to three decimal places as needed.) (d) which of the events can be considered unusual? explain. select all that apply. the event in part left parenthesis a right parenthesis is unusual because its probability is less than or equal to 0.05. the event in part (b) is unusual because its probability is less than or equal to 0.05. none of these events are unusual. the event in part (c) is unusual because its probability is less than or equal to 0.05.

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