Question

A husband and wife, Ed and Rina, share a digital music player that has a feature that randomly selects which song to play. A total of 3476 songs have been loaded into the player, some by Ed and the rest by Rina. They are interested in determining whether they have each loaded different proportions of songs into the player. Suppose that when the player was in the random-selection mode, 38 of the first 58 songs selected were songs loaded by Rina. Let p denote the proportion of songs that were loaded by Rina.

(a) State the null and alternative hypotheses to be tested. How strong is the evidence that Ed and Rina have each loaded a different proportion of songs into the player? Make sure to check the conditions for the use of this test. (Round your test statistic to two decimal places and your P-value to three decimal places. Assume a 95% confidence level.)

z = 2.36

P-value = 0.018

Conclusion: There is strong evidence that the proportion of songs downloaded by Ed and Rina differs from 0.5.

(b) Are the conditions for the use of the large sample confidence interval met? Yes, the conditions are met.

If so, estimate with 95% confidence the proportion of songs that were loaded by Rina. Round your answers to 3 decimal places. _____ to ________

Answer 1

Solution :

This is the two tailed test .

The null and alternative hypothesis is ,

H₀ : p = 0.5

H_a : p 0.5

n = 58

x = 38

= x / n = 38 / 58 = 0.6552

P₀ = 0.5

1 - P₀ = 1 - 0.5 = 0.5

z = - P₀ / [P_{0 *} (1 - P₀ ) / n]

= 0.6552 - 0.5 / [(0.5 * 0.5) / 58]

= 2.364

Test statistic = 2.36

P(z > 2.36) = 1 - P(z < 2.36) = 1 - 0.991 = 0.009

P-value = 2 * 0.009 = 0.018

= 0.05

P-value <

Reject the null hypothesis .

There is strong evidence that the proportion of songs downloaded by Ed and Rina differs from 0.5.

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.96

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.96 * (((0.655 * 0.345) / 58)

= 0.122

A 95% confidence interval for population proportion p is ,

- E < P < + E

0.655 - 0.122 < p < 0.655 + 0.122

0.533 < p < 0.777