**Hypothesis Testing for Dream Patterns in Different Age Groups**A study was carried out to determine the proportion of individuals who dream in black and white versus color. The sample data includes:- 321 people over the age of 55, out of which 77 dream in black and white.- 311 people under the age of 25, of whom 13 dream in black and white.The aim is to test the hypothesis at a 0.01 significance level that the proportion of people over 55 who dream in black and white is greater than that of people under 25.**Step a: Hypothesis Test**We will test the claim using the following hypotheses:- **Null Hypothesis (H₀):** The proportions are equal, \( p_1 = p_2 \)- **Alternative Hypothesis (H₁):** The proportion for those over 55 is greater, \( p_1 > p_2 \)Select the appropriate option for these hypotheses:- Option A: \( H₀: p_1 = p_2 \), \( H₁: p_1 > p_2 \)**Identify the Test Statistic:**Calculate the test statistic \( z \) and round to two decimal places as required.

The question is about hypo. testing Given : No. of people over age 55 under study ( n1 ) = 321 No. of people who dream black and white from n1 ( x1 ) = 77 No. of people under the age 25 in study ( n2 ) = 311 No. of people who dream black and white from n2 ( x2 ) = 13 Level of signif. ( α ) = 0.01 To find : a ) Test the given claim using hypo. testinga ) Null and alt. hypo : H0 : p1 = p2H1 : p1 > p2 Test stat : z = p^1 - p^2 - p1 - p2 SEWhere,SE = p^ * ( 1 - p^ ) * 1n1 + 1n2p^ = x1 + x2n1 + n2p^1 = x1n1p^2 = x2n2 Plugging values in above formulae p^1 = 77321 = 0.239875p^2 = 13311 = 0.041801p^ = 77 + 13321 + 311 = 0.142405SE = 0.142405 * ( 1 - 0.142405 ) * 1321 + 1311 = 0.027805z = ( 0.239875 - 0.041801 ) - 00.027805z = 7.123599 Rounding value to 2 dec. place as z = 7.12 p value approach : Since alt. hypo. has > sign, so it is right tailed test p value = P ( Z > zstat ) = P ( Z > 7.12 ) = 1 - P ( Z ≤ 7.12 ) Using excel command, =NORMSDIST ( z score ) So, p value = 5.39679E-13 p value = 0.000000000000539679 p value ≈ 0 Decision rule : If p value < level of signif. ( α ) then reject H0 otherwise do not reject H0 Here, p value < α Hence, reject H0 Conclusion : There is suffi. evid. to support the claim that the prop. of people over age 55 who dream B and W is greater than the prop. of people under age 25 who dream B and W