A toy store chain claims that at least 80% of girls under 8 years old prefer dolls over other types of toys. After observing the buying pattern of many girls under 8 years old, we feel that this claim is inflated. In an attempt to dispose of this claim, we observe the buying pattern of 20 randomly selected girls under 8 years old, and we observe X, the number of girls who buy stuffed toys or dolls. We wish to test the hypothesis Ho : p = 0.8 against Ha : p < 0.8. Suppose we decide to accept the Ho if X > 12 (that is X ≥ 13). This means that if {X ≤ 12} (that is X < 13) we will reject Ho. (a) Find α(Type I error). (b) Find β(Type II error) for p = 0.6. (c) Find β for p = 0.4.(d) Find the rejection region of the form {X ≤ K} so that (i) α = 0.01; (ii) α = 0.05.(e) For the alternative Ha: p = 0.6, find β for the values of α in part (d).
A toy store chain claims that at least 80% of girls under 8 years old prefer dolls over other types of toys. After observing the buying pattern of many girls under 8 years old, we feel that this claim is inflated. In an attempt to dispose of this claim, we observe the buying pattern of 20 randomly selected girls under 8 years old, and we observe X, the number of girls who buy stuffed toys or dolls. We wish to test the hypothesis Ho : p = 0.8 against Ha : p < 0.8. Suppose we decide to accept the Ho if X > 12 (that is X ≥ 13). This means that if {X ≤ 12} (that is X < 13) we will reject Ho. (a) Find α(Type I error). (b) Find β(Type II error) for p = 0.6. (c) Find β for p = 0.4.(d) Find the rejection region of the form {X ≤ K} so that (i) α = 0.01; (ii) α = 0.05.(e) For the alternative Ha: p = 0.6, find β for the values of α in part (d).
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