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	628	2,301

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 average 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 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.

• Part (d)

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

  P'₁ − P'₂

  ~  ? B H P Exp N   ,  

• 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.)
  ---Select--- z t =

• 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 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 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 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 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.

• 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.)
  α =
  
  (ii) Decision:

  reject the null hypothesisdo not reject the null hypothesis    

  
  (iii) Reason for decision:

  Since p-value < α, we reject the null hypothesis.Since p-value < α, we do not reject the null hypothesis.    Since p-value > α, we reject the null hypothesis.Since p-value > α, we do not 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 t-distribution will be used because the samples are dependent.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 standard normal distribution will be used because the samples involve the difference in proportions.

Answer 1

For sample 1, we have that the sample size is N_1= 628,

the number of favorable cases is X_1 = 69,

so then the sample proportion is

For sample 2, we have that the sample size is N_2 = 2301,

the number of favorable cases is X_2 = 231,

so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.01.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p_1 = p_2

Ha: p1​ > p2​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.01,

and the critical value for a right-tailed test is z_c = 2.33

The rejection region for this right-tailed test is

R={z:z>2.33}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that

Z = 0.695 ≤ zc​ =2.33,

it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is p = 0.2437,

and since p = 0.2437 ≥ 0.01,

it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected.

Therefore, there is not enough evidence to claim that the population proportion p_1 is greater than p_2​, at the 0.01 significance level.

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