Question

Automobiles purchased: An automobile owner found that 20 years ago, 76% of Americans said that they would prefer to purchase an American automobile. He believes that the number is much less than 76% today. He selected a random sample of 5 Americans and found that 38 said that they would prefer an American automobile. Can it be concluded that the percentagetoday is less than 76%? At α = 0.01, is he correct?

Answer 1

SOLUTION:

From given data,

Automobiles purchased: An automobile owner found that 20 years ago, 76% of Americans said that they would prefer to purchase an American automobile. He believes that the number is much less than 76% today. He selected a random sample of 50 Americans and found that 38 said that they would prefer an American automobile. Can it be concluded that the percentage today is less than 76%? At α = 0.01, is he correct?

Here we have given that,

Claim: To check whether the American automobile prefer Americans to purchase number is much less than 74 % or not.

The Hypothesis is

Ho: P =0.74 ( Null hypothesis)

v/s

H₁: P < 0.74 ( Alternative hypothesis)

Now,

n=number of observation = 50

x= number of Americans prefer automobiles=38

Now, we estimate the proportion $\hat{p}$ as

$\hat{p}$ =sample proportion = x / n = 38/50 = 0.76

Test statistic:

Z = ( $\hat{p}$ - P) /

VP(1-p) /n

Z = (  0.76 - 0.74) /

уб.74(1-074)/5

Z = 0.02 / 0.196163197

Z =0.10

Now we find the P-value

this is one tailed test

α = level of significance= 0.01

P-value=P( Z < 0.10)

= 0.53983   (using z standard normal table)

Decision:

Here P-value > 0.01 \alpha

That is here we fail to reject Ho(Null Hypothesis)

Conclusion:

there is not sufficient evidence that the American automobile prefer Americans to purchase number is much less than 76 %

dear ma'm /sir ...in your question you given as   "random sample of 5 Americans" but with that value we won't get the answer...as per my knowledge i was taken as   "random sample of 50 Americans"...if you want any changes please comment me.

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