Two computer users were discussing tablet computers. A higher proportion of people ages 16 to 29 use tablets than the proportion of people age 30 and older. The table below details the number of tablet owners for each age group. Test at the 1% level of significance. (For subscripts let 1 = 16-29 year old users, and 2 = 30 years old and older users.)

16–29 year olds 30 years old and older

Own a Tablet 69 231

Sample Size 625 2,313

NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Part (a)

State the null hypothesis.

H₀: p₁ < p₂

H₀: p₁ > p₂

H₀: p₁ ≠ p₂

H₀: p₁ = p₂

H₀: p₁ ≥ p₂

Part (b)

State the alternative hypothesis.

H_a: p₁ < p₂

H_a: p₁ ≠ p₂

H_a: p₁ = p₂

H_a: p₁ > p₂

H_a: p₁ ≤ p₂

Part (c)

In words, state what your random variable
P'₁ − P'₂
represents.

P'₁ − P'₂
represents the difference between the average proportions of 16-29 year old tablet users from that of the 30 years old and older tablet users.
P'₁ − P'₂
represents the difference between the average numbers of 16-29 year old tablet users from that of the 30 years old and older tablet users.
P'₁ − P'₂
represents the difference between the proportions of 16-29 year old tablet users from that of the 30 years old and older tablet users.
P'₁ − P'₂
represents the average difference between the proportions of 16-29 year old tablet users from that of the 30 years old and older tablet users.

Part (d)

State the distribution to use for the test. (Round your answers to four decimal places.)

P'₁ − P'₂
~

  ,

Part (e)

What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.)
    =

Part (f)

What is the p-value? (Round your answer to four decimal places.)

Explain what the p-value means for this problem.
If
H₀
is false, then there is a chance equal to the p-value that the proportion of tablet users that are 16-29 year old users is 0.01 less than the proportion of tablet users that are 30 years old and older tablet users.If
H₀
is true, then there is a chance equal to the p-value that the proportion of tablet users that are 16-29 year old users is at least 0.01 less than the proportion of tablet users that are 30 years old and older tablet users.    If
H₀
is false, then there is a chance equal to the p-value that the proportion of tablet users that are 16-29 year old users is at least 0.01 more than the proportion of tablet users that are 30 years old and older tablet users.If
H₀
is true, then there is a chance equal to the p-value that the proportion of tablet users that are 16-29 year old users is 0.01 more than the proportion of tablet users that are 30 years old and older tablet users.

Part (g)

Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value.

Part (h)

Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.
(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
? =

(ii) Decision:
reject the null hypothesisdo not reject the null hypothesis

(iii) Reason for decision:
Since p-value > ?, we do not reject the null hypothesis.Since p-value < ?, we do not reject the null hypothesis.    Since p-value > ?, we reject the null hypothesis.Since p-value < ?, we reject the null hypothesis.

(iv) Conclusion:
There is sufficient evidence to conclude that a higher proportion of tablet owners are aged 16 to 29 years old than are 30 years old and older.There is not sufficient evidence to conclude that a higher proportion of tablet owners are aged 16 to 29 years old than are 30 years old and older.

Part (i)

Explain how you determined which distribution to use.
The standard normal distribution will be used because the samples are independent and the population standard deviation is known.The t-distribution will be used because the samples are independent and the population standard deviation is not known.    The t-distribution will be used because the samples are dependent.The standard normal distribution will be used because the samples involve the difference in proportions.

Question

Two computer users were discussing tablet computers. A higher proportion of people ages 16 to 29 use tablets than the proportion of people age 30 and older. The table below details the number of tablet owners for each age group. Test at the 1% level of significance. (For subscripts let 1 = 16-29 year old users, and 2 = 30 years old and older users.)

	16–29 year olds	30 years old and older
Own a Tablet	69	231
Sample Size	625	2,313

NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Part (a)

State the null hypothesis.

H₀: p₁ < p₂

H₀: p₁ > p₂

H₀: p₁ ≠ p₂

H₀: p₁ = p₂

H₀: p₁ ≥ p₂
Part (b)

State the alternative hypothesis.

H_a: p₁ < p₂

H_a: p₁ ≠ p₂

H_a: p₁ = p₂

H_a: p₁ > p₂

H_a: p₁ ≤ p₂
Part (c)

In words, state what your random variable
P'₁ − P'₂
represents.

P'₁ − P'₂
represents the difference between the average proportions of 16-29 year old tablet users from that of the 30 years old and older tablet users.
P'₁ − P'₂
represents the difference between the average numbers of 16-29 year old tablet users from that of the 30 years old and older tablet users.
P'₁ − P'₂
represents the difference between the proportions of 16-29 year old tablet users from that of the 30 years old and older tablet users.
P'₁ − P'₂
represents the average difference between the proportions of 16-29 year old tablet users from that of the 30 years old and older tablet users.
Part (d)

State the distribution to use for the test. (Round your answers to four decimal places.)

P'₁ − P'₂
~

,
Part (e)

What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.)
=
Part (f)

What is the p-value? (Round your answer to four decimal places.)

Explain what the p-value means for this problem.
If
H₀
is false, then there is a chance equal to the p-value that the proportion of tablet users that are 16-29 year old users is 0.01 less than the proportion of tablet users that are 30 years old and older tablet users.If
H₀
is true, then there is a chance equal to the p-value that the proportion of tablet users that are 16-29 year old users is at least 0.01 less than the proportion of tablet users that are 30 years old and older tablet users. If
H₀
is false, then there is a chance equal to the p-value that the proportion of tablet users that are 16-29 year old users is at least 0.01 more than the proportion of tablet users that are 30 years old and older tablet users.If
H₀
is true, then there is a chance equal to the p-value that the proportion of tablet users that are 16-29 year old users is 0.01 more than the proportion of tablet users that are 30 years old and older tablet users.
Part (g)

Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
Part (h)

Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.
(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
? =

(ii) Decision:
reject the null hypothesisdo not reject the null hypothesis

(iii) Reason for decision:
Since p-value > ?, we do not reject the null hypothesis.Since p-value < ?, we do not reject the null hypothesis.    Since p-value > ?, we reject the null hypothesis.Since p-value < ?, we reject the null hypothesis.

(iv) Conclusion:
There is sufficient evidence to conclude that a higher proportion of tablet owners are aged 16 to 29 years old than are 30 years old and older.There is not sufficient evidence to conclude that a higher proportion of tablet owners are aged 16 to 29 years old than are 30 years old and older.
Part (i)

Explain how you determined which distribution to use.
The standard normal distribution will be used because the samples are independent and the population standard deviation is known.The t-distribution will be used because the samples are independent and the population standard deviation is not known. The t-distribution will be used because the samples are dependent.The standard normal distribution will be used because the samples involve the difference in proportions.

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