Math / Data & Statistics

QuestionIt is known that roughly 2/3 of all human beings have a dominant right foot or eye. Is there also right-sided dominance in kissing behavior? An article reported that in a random sample of 115 kissing couples, both people in 73 of the couples tended to lean more to the right than to the left. (Use a = 0.05.) USE SALT (a) If 2/3 of all kissing couples exhibit this right-leaning behavior, what is the probability that the number in a sample of 115 who do so differs from the expected value by at least as much as what was actually observed? (Round your answer to four decimal places.) (b) Does the result of the experiment suggest that the 2/3 figure is implausible for kissing behavior? State the appropriate null and alternative hypotheses. Ho: P = 2/3 Hp < 2/3 Ho P = 2/3 H: p + 2/3 O Ho: p = 2/3 HP ≤ 2/3 OHP = 2/3 H:p> 2/3 Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) == P-value =

Studdy Solution

STEP 1

What is this asking? We're checking if the idea that 2/3 of couples lean right when kissing is a reasonable idea, given that 73 out of 115 couples in a study actually leaned right. Watch out! Don't mix up the expected number of right-leaning couples with the observed number.
Also, remember that a small p-value means we have evidence against the 2/3 hypothesis!

STEP 2

1. Calculate Expected Value and Standard Deviation
2. Calculate the Z-score
3. Calculate the p-value
4. Interpret the p-value

STEP 3

If the 2/3 hypothesis is true, we'd expect $(2/3) \cdot 115 \approx 76.67$ couples to lean right.
This is our expected value.

STEP 4

The standard deviation tells us how much variation we expect around the mean.
It's calculated as $\sqrt{n \cdot p \cdot (1-p)}$ , where $n$ is the sample size ( $n = \textbf{115}$ ) and $p$ is the hypothesized proportion ( $p = \textbf{2/3}$ ).
So, our standard deviation is $\sqrt{115 \cdot (2/3) \cdot (1/3)} \approx \sqrt{25.56} \approx \textbf{5.06}$ .

STEP 5

We observed 73 right-leaning couples, and we expected about 76.67.
The difference is $73 - 76.67 = \textbf{-3.67}$ .

STEP 6

The Z-score tells us how many standard deviations away from the expected value our observation is.
It's calculated as $\frac{\text{observed} - \text{expected}}{\text{standard deviation}}$ .
So, our Z-score is $\frac{-3.67}{5.06} \approx \textbf{-0.72}$ .

STEP 7

The p-value is the probability of observing a result as extreme as ours (or more extreme) if the 2/3 hypothesis is actually true.

STEP 8

Since we're looking at deviations in both directions (more or less than the expected value), we want a two-tailed p-value.
Looking up our Z-score of $\textbf{-0.72}$ in a Z-table (or using a calculator), we find a p-value of approximately $\textbf{0.4716}$ .

STEP 9

Our p-value ( $\textbf{0.4716}$ ) is much larger than our significance level ( $\alpha = \textbf{0.05}$ ).

STEP 10

A large p-value means we don't have enough evidence to reject the 2/3 hypothesis.
It seems plausible that 2/3 of kissing couples lean right!

STEP 11

$P$ -value = 0.4716.
The result of the experiment does not suggest that the 2/3 figure is implausible for kissing behavior.
The appropriate hypotheses are $H_0: p = 2/3$ and $H_a: p \ne 2/3$ .
The test statistic is approximately -0.72.

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Studdy Solution

STEP 1

STEP 2

1. Calculate Expected Value and Standard Deviation
2. Calculate the Z-score
3. Calculate the p-value
4. Interpret the p-value

STEP 3

If the 2/3 hypothesis is true, we'd expect $(2/3) \cdot 115 \approx 76.67$ couples to lean right.
This is our expected value.

STEP 4

STEP 5

We observed 73 right-leaning couples, and we expected about 76.67.
The difference is $73 - 76.67 = \textbf{-3.67}$ .

STEP 6

The Z-score tells us how many standard deviations away from the expected value our observation is.
It's calculated as $\frac{\text{observed} - \text{expected}}{\text{standard deviation}}$ .
So, our Z-score is $\frac{-3.67}{5.06} \approx \textbf{-0.72}$ .

STEP 7

The p-value is the probability of observing a result as extreme as ours (or more extreme) if the 2/3 hypothesis is actually true.

STEP 8

Since we're looking at deviations in both directions (more or less than the expected value), we want a two-tailed p-value.
Looking up our Z-score of $\textbf{-0.72}$ in a Z-table (or using a calculator), we find a p-value of approximately $\textbf{0.4716}$ .

STEP 9

Our p-value ( $\textbf{0.4716}$ ) is much larger than our significance level ( $\alpha = \textbf{0.05}$ ).

STEP 10

A large p-value means we don't have enough evidence to reject the 2/3 hypothesis.
It seems plausible that 2/3 of kissing couples lean right!

STEP 11

$P$ -value = 0.4716.
The result of the experiment does not suggest that the 2/3 figure is implausible for kissing behavior.
The appropriate hypotheses are $H_0: p = 2/3$ and $H_a: p \ne 2/3$ .
The test statistic is approximately -0.72.

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Studdy Solution

STEP 1

STEP 2

1. Calculate Expected Value and Standard Deviation2. Calculate the Z-score3. Calculate the p-value4. Interpret the p-value

STEP 3

If the 2/3 hypothesis is true, we'd *expect* (2/3)⋅115≈76.67(2/3) \cdot 115 \approx 76.67(2/3)⋅115≈76.67 couples to lean right.This is our **expected value**.

STEP 4

STEP 5

We observed **73** right-leaning couples, and we expected about **76.67**.The difference is 73−76.67=-3.6773 - 76.67 = \textbf{-3.67}73−76.67=-3.67.

STEP 6

STEP 7

The **p-value** is the probability of observing a result as extreme as ours (or *more* extreme) *if* the 2/3 hypothesis is actually true.

STEP 8

Since we're looking at deviations in *both* directions (more *or* less than the expected value), we want a two-tailed p-value.Looking up our Z-score of -0.72\textbf{-0.72}-0.72 in a Z-table (or using a calculator), we find a p-value of approximately 0.4716\textbf{0.4716}0.4716.

STEP 9

Our p-value (0.4716\textbf{0.4716}0.4716) is much larger than our significance level (α=0.05\alpha = \textbf{0.05}α=0.05).

STEP 10

A large p-value means we *don't* have enough evidence to reject the 2/3 hypothesis.It seems plausible that 2/3 of kissing couples lean right!

STEP 11

PPP-value = 0.4716.The result of the experiment does *not* suggest that the 2/3 figure is implausible for kissing behavior.The appropriate hypotheses are H0:p=2/3H_0: p = 2/3H0​:p=2/3 and Ha:p≠2/3H_a: p \ne 2/3Ha​:p=2/3.The test statistic is approximately -0.72.

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1. Calculate Expected Value and Standard Deviation
2. Calculate the Z-score
3. Calculate the p-value
4. Interpret the p-value

If the 2/3 hypothesis is true, we'd expect $(2/3) \cdot 115 \approx 76.67$ couples to lean right.
This is our expected value.

We observed 73 right-leaning couples, and we expected about 76.67.
The difference is $73 - 76.67 = \textbf{-3.67}$ .

The p-value is the probability of observing a result as extreme as ours (or more extreme) if the 2/3 hypothesis is actually true.

Since we're looking at deviations in both directions (more or less than the expected value), we want a two-tailed p-value.
Looking up our Z-score of $\textbf{-0.72}$ in a Z-table (or using a calculator), we find a p-value of approximately $\textbf{0.4716}$ .

Our p-value ( $\textbf{0.4716}$ ) is much larger than our significance level ( $\alpha = \textbf{0.05}$ ).

A large p-value means we don't have enough evidence to reject the 2/3 hypothesis.
It seems plausible that 2/3 of kissing couples lean right!

$P$ -value = 0.4716.
The result of the experiment does not suggest that the 2/3 figure is implausible for kissing behavior.
The appropriate hypotheses are $H_0: p = 2/3$ and $H_a: p \ne 2/3$ .
The test statistic is approximately -0.72.